IV Quarter Exam Contents: Temperature and Thermal Equilibrium Defining Temperature Measuring Temperature Temperature Conversion Electric Charge Properties of Electric Charge Transfer of Electric Charge Electric Current and Resistance Electric Power

Contents

Month:

Magnetic Phenomena Magnets and Magnetic Fields

Magnetism from Electricity

Magnetic Forces

Types of Magnets

The object which contains magnetic field is called a magnet. It has ability to attract iron, nickel etc. it was first discovered by Greek when iron is attracted by lodestone but now days, it was discovered as magnets. These are also can be formed by artificial methods in different size and shape according to their requirement. The commonly used magnet is bar magnet which is a rectangular bar in long shape which attracts ferrous objects.

Similarly, compass needle is also magnet which can freely move in a horizontal direction on a pivot with one end is in North direction and the other end is in the South direction. The surrounding space of magnet feels the magnetic field. It is due to the magnetic force around the magnet. So, when we place a bar magnet in a field then it will be in magnetic forces which is continuing even after removing it from magnetic field. The magnetic field is produced by the flow of current. There are different types of magnets which are based on their magnetic properties and their efficiency to show magnetism.

Permanent Magnet

These are types of magnets which retain certain degree of magnetism once they are magnetized. That is why they are called as permanent magnets. Some of the natural magnets which are the examples of Permanent magnets include lodestone or magnetite. A magnet always have two poles i.e. north and south poles. Suppose that we have a magnet whose length is 100 cm and it has two poles. Now if we divide this 100 cm magnet into two halves of 50 cm each, we will get two magnets each having north and south poles. If we divide this 100 cm magnet into four halves then we will get four magnets each having two poles. Therefore,a magnet with single pole never exists in nature.

The Earth's magnetic

field appears

to come from

a giant bar

magnet, but

with its south pole located

up near the

Earth's north pole

(near Canada). The magnetic field lines come out of the Earth

near Antarctica and enter near Canada.

Electric Phenomena
Electric Circuits Current and Resistance Electric Power

Electric Field and the Movement of Charge

Perhaps one of the most useful yet taken-for-granted accomplishments of the recent centuries is the development of electric circuits. The flow of charge through wires allows us to cook our food, light our homes, air-condition our work and living space, entertain us with movies and music and even allows us to drive to work or school safely. In this unit of The Physics Classroom, we will explore the reasons for why charge flows through wires of electric circuits and the variables that affect the rate at which it flows. The means by which moving charge delivers electrical energy to appliances in order to operate them will be discussed in detail. One of the fundamental principles that must be understood in order to grasp electric circuits pertains to the concept of how an electric field can influence charge within a circuit as it moves from one location to another. The concept of electric fieldwas first introduced in the unit on Static Electricity. In that unit, electric force was described as a non-contact force. A charged balloon can have an attractive effect upon an oppositely charged balloon even when they are not in contact. The electric force acts over the distance separating the two objects. Electric force is an action-at-a-distance force.

Action-at-a-distance forces are sometimes referred to as field forces. The concept of a field force is utilized by scientists to explain this rather unusual force phenomenon that occurs in the absence of physical contact. The space surrounding a charged object is affected by the presence of the charge; an electric field is established in that space. A charged object creates an electric field - an alteration of the space or field in the region that surrounds it. Other charges in that field would feel the unusual alteration of the space. Whether a charged object enters that space or not, the electric field exists. Space is altered by the presence of a charged object; other objects in that space experience the strange and mysterious qualities of the space. As another charged object enters the space and moves deeper and deeper into the field, the effect of the field becomes more and more noticeable.

Electric field is a vector quantity whose direction is defined as the direction that a positive test charge would be pushed when placed in the field. Thus, the electric field direction about a positive source charge is always directed away from the positive source. And the electric field direction about a negative source charge is always directed toward the negative source.

Measuring Temperature
Heat and Energy
Thermal Conduction Heat and Work

Temperature and Thermometers

What is Temperature?

Despite our built-in feel for temperature, it remains one of those concepts in science that is difficult to define. It seems that a tutorial page exploring the topic of temperature and thermometers should begin with a simple definition of temperature. But it is at this point that I'm stumped. So I turn to that familiar resource, Dictionary.com ... where I find definitions that vary from the simple-yet-not-too-enlightening to the too-complex-to-be-enlightening. At the risk of doing a belly flop in the pool of enlightenment, I will list some of those definitions here:

The degree of hotness or coldness of a body or environment.

A measure of the warmth or coldness of an object or substance with reference to some standard value.

A measure of the average kinetic energy of the particles in a sample of matter, expressed in terms of units or degrees designated on a standard scale.

A measure of the ability of a substance, or more generally of any physical system, to transfer heat energy to another physical system.

Any of various standardized numerical measures of this ability, such as the Kelvin, Fahrenheit, and Celsius scale.

For certain, we are comfortable with the first two definitions - the degree or measure of how hot or cold and object is. But our understanding of temperature is not furthered by such definitions. The third and the fourth definitions that reference the kinetic energy of particles and the ability of a substance to transfer heat are scientifically accurate. However, these definitions are far too sophisticated to serve as good starting points for a discussion of temperature. So we will resign to a definition similar to the fifth one that is listed - temperature can be defined as the reading on a thermometer. Admittedly, this definition lacks the power that is needed for eliciting the much-desired Aha! Now I Understand! moment. Nonetheless it serves as a great starting point for this lesson and heat and temperature.Temperatureis what the thermometer reads. Whatever it is that temperature is a measure of, it is reflected by the reading on a thermometer.

How a Thermometer Works

Today, there are a variety of types of thermometers. The type that most of us are familiar with from science class is the type that consists of a liquid encased in a narrow glass column. Older thermometers of this type used liquid mercury. In response to our understanding of the health concerns associated with mercury exposure, these types of thermometers usually use some type of liquid alcohol. These liquid thermometers are based on the principal of thermal expansion. When a substance gets hotter, it expands to a greater volume. Nearly all substances exhibit this behavior of thermal expansion. It is the basis of the design and operation of thermometers.

As the temperature of the liquid in a thermometer increases, its volume increases. The liquid is enclosed in a tall, narrow glass (or plastic) column with a constant cross-sectional area. The increase in volume is thus due to a change in height of the liquid within the column. The increase in volume, and thus in the height of the liquid column, is proportional to the increase in temperature. Suppose that a 10-degree increase in temperature causes a 1-cm increase in the column's height. Then a 20-degree increase in temperature will cause a 2-cm increase in the column's height. And a 30-degree increase in temperature will cause s 3-cm increase in the column's height. The relationship between the temperature and the column's height is linear over the small temperature range for which the thermometer is used. This linear relationship makes the calibration of a thermometer a relatively easy task.

The calibration of any measuring tool involves the placement of divisions or marks upon the tool to measure a quantity accurately in comparison to known standards. Any measuring tool - even a meter stick - must be calibrated. The tool needs divisions or markings; for instance, a meter stick typically has markings every 1-cm apart or every 1-mm apart. These markings must be accurately placed and the accuracy of their placement can only be judged when comparing it to another object known to have an accurate length. A thermometer is calibrated by using two objects of known temperatures. The typical process involves using the freezing point and the boiling point of water. Water is known to freeze at 0°C and to boil at 100°C at an atmospheric pressure of 1 atm. By placing a thermometer in mixture of ice water and allowing the thermometer liquid to reach a stable height, the 0-degree mark can be placed upon the thermometer. Similarly, by placing the thermometer in boiling water (at 1 atm of pressure) and allowing the liquid level to reach a stable height, the 100-degree mark can be placed upon the thermometer. With these two markings placed upon the thermometer, 100 equally spaced divisions can be placed between them to represent the 1-degree marks. Since there is a linear relationship between the temperature and the height of the liquid, the divisions between 0 degree and 100 degree can be equally spaced. With a calibrated thermometer, accurate measurements can be made of the temperature of any object within the temperature range for which it has been calibrated.

Temperature Scales

The thermometer calibration process described above results in what is known as a centigrade thermometer. A centigrade thermometer has 100 divisions or intervals between the normal freezing point and the normal boiling point of water. Today, the centigrade scale is known as the Celsius scale, named after the Swedish astronomer Anders Celsius who is credited with its development. The Celsius scale is the most widely accepted temperature scale used throughout the world. It is the standard unit of temperature measurement in nearly all countries, the most notable exception being the United States. Using this scale, a temperature of 28 degrees Celsius is abbreviated as 28°C.

Traditionally slow to adopt the metric system and other accepted units of measurements, the United States more commonly uses the Fahrenheit temperature scale. A thermometer can be calibrated using the Fahrenheit scale in a similar manner as was described above. The difference is that the normal freezing point of water is designated as 32 degrees and the normal boiling point of water is designated as 212 degrees in the Fahrenheit scale. As such, there are 180 divisions or intervals between these two temperatures when using the Fahrenheit scale. The Fahrenheit scale is named in honor of German physicist Daniel Fahrenheit. A temperature of 76 degree Fahrenheit is abbreviated as 76°F. In most countries throughout the world, the Fahrenheit scale has been replaced by the use of the Celsius scale.

Temperatures expressed by the Fahrenheit scale can be converted to the Celsius scale equivalent using the equation below:

°C = (°F - 32°)/1.8

Similarly, temperatures expressed by the Celsius scale can be converted to the Fahrenheit scale equivalent using the equation below:

°F= 1.8•°C + 32°

The Kelvin Temperature Scale

While the Celsius and Fahrenheit scales are the most widely used temperature scales, there are several other scales that have been used throughout history. For example, there is the Rankine scale, the Newton scale and the Romer scale, all of which are rarely used. Finally, there is the Kelvin temperature scale, which is the standard metric system of temperature measurement and perhaps the most widely used temperature scale used among scientists. The Kelvin temperature scale is similar to the Celsius temperature scale in the sense that there are 100 equal degree increments between the normal freezing point and the normal boiling point of water. However, the zero-degree mark on the Kelvin temperature scale is 273.15 units cooler than it is on the Celsius scale. So a temperature of 0 Kelvin is equivalent to a temperature of -273.15 °C. Observe that the degree symbol is not used with this system. So a temperature of 300 units above 0 Kelvin is referred to as 300 Kelvin and not 300 degree Kelvin; such a temperature is abbreviated as 300 K. Conversions between Celsius temperatures and Kelvin temperatures (and vice versa) can be performed using one of the two equations below.

°C = K - 273.15°

K = °C + 273.15

The zero point on the Kelvin scale is known as absolute zero. It is the lowest temperature that can be achieved. The concept of an absolute temperature minimum was promoted by Scottish physicist William Thomson (a.k.a. Lord Kelvin) in 1848. Thomson theorized based on thermodynamic principles that the lowest temperature which could be achieved was -273°C. Prior to Thomson, experimentalists such as Robert Boyle (late 17th century) were well aware of the observation that the volume (and even the pressure) of a sample of gas was dependent upon its temperature. Measurements of the variations of pressure and volume with changes in the temperature could be made and plotted. Plots of volume vs. temperature (at constant pressure) and pressure vs. temperature (at constant volume) reflected the same conclusion - the volume and the pressure of a gas reduces to zero at a temperature of -273°C. Since these are the lowest values of volume and pressure that are possible, it is reasonable to conclude that -273°C was the lowest temperature that was possible.

Thomson referred to this minimum lowest temperature as absolute zero and argued that a temperature scale be adopted that had absolute zero as the lowest value on the scale. Today, that temperature scale bears his name. Scientists and engineers have been able to cool matter down to temperatures close to -273.15°C, but never below it. In the process of cooling matter to temperatures close to absolute zero, a variety of unusual properties have been observed. These properties include superconductivity, superfluidity and a state of matter known as a Bose-Einstein condensate.

Thermometers as Speedometers

Temperature is defined as the reading on a thermometer. The process of calibrating a thermometer was explained and the variety of commonly used temperature scales were described. Finally, the concept of an absolute lowest temperature was discussed. But in the end, the fundamental definition of temperature was not given. Temperature was only defined in practical terms - the reading on a thermometer. Now we have to answer the more fundamental question: what is the reading on a thermometer the reflection of? What is temperature a measure of?

It is at this point that we can use a more sophisticated definition of temperature. Temperature is a measure of the average kinetic energy of the particles within a sample of matter. Kinetic energy was defined as the energy of motion. An object ... or a particle ... that is moving has kinetic energy. There are three common forms of kinetic energy - vibrational kinetic energy, rotational kinetic energy and translational kinetic energy. Up to this point of the tutorial, we have associated kinetic energy with the movement of an object (or particle) from one location to another. This is referred to as translational kinetic energy. A ball moving through space has translational kinetic energy. An object can also have vibrational kinetic energy; this is the energy of motion possessed by an object that is oscillating or vibrating about a fixed position. A mass attached to a spring has vibrational kinetic energy. Such a mass is not permanently displaced from its position like a ball moving through space. Finally an object can have rotational kinetic energy; this is the energy associated with an object that is rotating about an imaginary axis of rotation. A spinning top isn't moving through space and isn't vibrating about a fixed position, but there is still kinetic energy associated with its motion about an axis of rotation. This form of kinetic energy is called rotational kinetic energy.

A sample of matter consists of particles that can be vibrating, rotating and moving through the space of its container. So at the particle level, a sample of matter possesses kinetic energy. A warm cup of water on a countertop may appear to be as still as can be; yet it still has kinetic energy. At the particle level, there are atoms and molecules that are vibrating, rotating and moving through the space of its container. Stick a thermometer in the cup of water and you will see the evidence that the water possesses kinetic energy. The water's temperature, as reflected by the thermometer's reading, is a measure of the average amount of kinetic energy possessed by the water molecules.

When the temperature of an object increases, the particles that compose the object begin to move faster. They either vibrate more rapidly, rotate with greater frequency or move through space with a greater speed. Increasing the temperature causes an increase in the particle speed. So as a sample of water in a pot is heated, its molecules begin to move with greater speed and this greater speed is reflected by a higher thermometer reading. Similarly, if a sample of water is placed in the freezer, its molecules begin to move slower (with a lower speed) and this is reflected by a lower thermometer reading. It is in this sense that a thermometer can be thought of as a speedometer.

What is Heat?

Consider a very hot mug of coffee on the countertop of your kitchen. For discussion purposes, we will say that the cup of coffee has a temperature of 80°C and that the surroundings (countertop, air in the kitchen, etc.) has a temperature of 26°C. What do you suppose will happen in this situation? I suspect that you know that the cup of coffee will gradually cool down over time. At 80°C, you wouldn't dare drink the coffee. Even the coffee mug will likely be too hot to touch. But over time, both the coffee mug and the coffee will cool down. Soon it will be at a drinkable temperature. And if you resist the temptation to drink the coffee, it will eventually reach room temperature. The coffee cools from 80°C to about 26°C. So what is happening over the course of time to cause the coffee to cool down? The answer to this question can be both macroscopic and particulate in nature.

On the macroscopic level, we would say that the coffee and the mug are transferring heat to the surroundings. This transfer of heat occurs from the hot coffee and hot mug to the surrounding air. The fact that the coffee lowers its temperature is a sign that the average kinetic energy of its particles is decreasing. The coffee is losing energy. The mug is also lowering its temperature; the average kinetic energy of its particles is also decreasing. The mug is also losing energy. The energy that is lost by the coffee and the mug is being transferred to the colder surroundings. We refer to this transfer of energy from the coffee and the mug to the surrounding air and countertop as heat. In this sense, heat is simply the transfer of energy from a hot object to a colder object.

Now let's consider a different scenario - that of a cold can of pop placed on the same kitchen counter. For discussion purposes, we will say that the pop and the can which contains it has a temperature of 5°C and that the surroundings (countertop, air in the kitchen, etc.) has a temperature of 26°C. What will happen to the cold can of pop over the course of time? Once more, I suspect that you know the answer. The cold pop and the container will both warm up to room temperature. But what is happening to cause these colder-than-room-temperature objects to increase their temperature? Is the cold escaping from the pop and its container? No! There is no such thing as the cold escaping orleaking. Rather, our explanation is very similar to the explanation used to explain why the coffee cools down. There is a heat transfer.

Over time, the pop and the container increase their temperature. The temperature rises from 5°C to nearly 26°C. This increase in temperature is a sign that the average kinetic energy of the particles within the pop and the container is increasing. In order for the particles within the pop and the container to increase their kinetic energy, they must be gaining energy from somewhere. But from where? Energy is being transferred from the surroundings (countertop, air in the kitchen, etc.) in the form of heat. Just as in the case of the cooling coffee mug, energy is being transferred from the higher temperature objects to the lower temperature object. Once more, this is known as heat - the transfer of energy from the higher temperature object to a lower temperature object.

Both of these scenarios could be summarized by two simple statements. An object decreases its temperature by releasing energy in the form of heat to its surroundings. And an object increases its temperature by gaining energy in the form of heat from its surroundings. Both the warming upand the cooling down of objects works in the same way - by heat transfer from the higher temperature object to the lower temperature object. So now we can meaningfully re-state the definition of temperature. Temperature is a measure of the ability of a substance, or more generally of any physical system, to transfer heat energy to another physical system. The higher the temperature of an object is, the greater the tendency of that object to transfer heat. The lower the temperature of an object is, the greater the tendency of that object to be on the receiving end of the heat transfer.

But perhaps you have been asking: what happens to the temperature of surroundings? Do the countertop and the air in the kitchen increase their temperature when the mug and the coffee cool down? And do the countertop and the air in the kitchen decrease its temperature when the can and its pop warm up? The answer is a resounding Yes! The proof? Just touch the countertop - it should feel cooler or warmer than before the coffee mug or pop can were placed on the countertop. But what about the air in the kitchen? Now that's a little more difficult to present a convincing proof of. The fact that the volume of air in the room is so large and that the energy quickly diffuses away from the surface of the mug and of the means that the temperature change of the air in the kitchen will be abnormally small. In fact, it will be negligibly small. There would have to be a lot more heat transfer before there is a noticeable temperature change.

Thermal Equilibrium

In the discussion of the cooling of the coffee mug, the countertop and the air in the kitchen were referred to as thesurroundings. It is common in physics discussions of this type to use a mental framework of a system and thesurroundings. The coffee mug (and the coffee) would be regarded as the system and everything else in the universe would be regarded as the surroundings. To keep it simple, we often narrow the scope of the surroundings from the rest of the universe to simply those objects that are immediately surrounding the system. This approach of analyzing a situation in terms of system and surroundings is so useful that we will adopt the approach for the rest of this chapter and the next.

Now let's imagine a third situation. Suppose that a small metal cup of hot water is placed inside of a larger Styrofoam cup of cold water. Let's suppose that the temperature of the hot water is initially 70°C and that the temperature of the cold water in the outer cup is initially 5°C. And let's suppose that both cups are equipped with thermometers (or temperature probes) that measure the temperature of the water in each cup over the course of time. What do you suppose will happen? Before you read on, think about the question and commit to some form of answer. When the cold water is done warming and the hot water is done cooling, will their temperatures be the same or different? Will the cold water warm up to a lower temperature than the temperature that the hot water cools down to? Or as the warming and cooling occurs, will their temperatures cross each other?

As you can see from the graph, the hot water cooled down to approximately 30°C and the cold water warmed up to approximately the same temperature. Heat is transferred from the high temperature object (inner can of hot water) to the low temperature object (outer can of cold water). If we designate the inner cup of hot water as the system, then we can say that there is a flow of water from the system to the surroundings. As long as there is a temperature difference between the system and the surroundings, there is a heat flow between them. The heat flow is more rapid at first as depicted by the steeper slopes of the lines. Over time, the temperature difference between system and surroundings decreases and the rate of heat transfer decreases. This is denoted by the gentler slope of the two lines. (Detailed information about rates of heat transfer will be discussed later in this lesson.) Eventually, the system and the surroundings reach the same temperature and the heat transfer ceases. It is at this point, that the two objects are said to have reached thermal equilibrium.

Heat Changes the Temperature of Objects

What does heat do? First, it changes the temperature of an object. If heat is transferred from an object to the surroundings, then the object can cool down and the surroundings can warm up. When heat is transferred to an object by its surroundings, then the object can warm up and the surroundings can cool down. Heat, once absorbed as energy, contributes to the overall internal energy of the object. One form of this internal energy is kinetic energy; the particles begin to move faster, resulting in a greater kinetic energy. This more vigorous motion of particles is reflected by a temperature increase. The reverse logic applies as well. Energy, once released as heat, results in a decrease in the overall internal energy of the object. Since kinetic energy is one of the forms of internal energy, the release of heat from an object causes a decrease in the average kinetic energy of its particles. This means that the particles move more sluggishly and the temperature of the object decreases. The release or absorption of energy in the form heat by an object is often associated with a temperature change of that object. This was the focus of the Thermometers as Speedometers. What can be said of the object can also be said of the surroundings. The release or absorption of energy in the form heat by the surroundings is often associated with a temperature change of the surroundings. We often find that the transfer of heat causes a temperature change in both system and surroundings. One warms up and the other cools down.

Heat Changes the State of Matter

But does the absorption or release of energy in the form of heat always cause a temperature change? Surprisingly, the answer is no. To illustrate why, consider the following situation, which is often demonstrated or even experimented with in a thermal physics unit in school. Para-dichlorobenzene, the main ingredient in many forms of mothballs, has a melting point of about 54 °C. Suppose that a sample of the chemical is collected in a test tube and heated to about 80°C. The para-dichlorobenzene will be in the liquid state (though much of it will have sublimed and be filling the room with a most noticeable aroma). Now suppose that a thermometer is inserted in the test tube and that the test tube is placed in a beaker of room temperature water. Temperature-time data can be collected every 10 seconds. Quite expectedly, one notices that the temperature of the para-dichlorobenzene gradually decreases. As heat is transferred from the high temperature test tube to the low temperature water, the temperature of the liquid para-dichlorbenzene decreases. But then quite unexpectedly, one would notice that this steady decrease in temperature ceases at about 54°C. Once the temperature of liquid para-dichlorbenzene decreases to 54°C, the thermometer level suddenly stands still. Based on the thermometer reading, you might think that no heat was being transferred. But a look in the test tube reveals dramatic change taking place. The liquid para-dichlorbenzene is crystallizing to form solid para-dichlorbenzene. Once the last trace of liquid para-dichlorbenzene vanishes (and it is in all solid form), the temperature begins to decrease again from 54°C to the temperature of the water. How can these observations help us to understand the question of what does heat do?

First, the decrease in temperature from 80°C to 54°C is easy to explain. Heat is transferred between two adjacent objects that are at different temperatures. The test tube and the para-dichlorbenzene are at a higher temperature than the surrounding water of the beaker. Heat will flow from the test tube of para-dichlorbenzene to the water, causing the para-dichlorbenzene to cool down and the water to warm up. And the decrease in temperature from 54°C to the temperature of the water in the beaker is also easily explainable. Two adjacent objects of different temperatures will transfer heat between them until thermal equilibrium is reached. The difficult explanation involves explaining what happens at 54°C. Why does the temperature no longer decrease when the liquid para-dichlorbenzene begins to crystallize? Is there still a transfer of heat between the test tube of para-dichlorbenzene and the beaker of water even when the temperature isn't changing?

The answer to the question Is heat being transferred? is a resounding yes! After all, the principle is that heat is always transferred between two adjacent objects that are at different temperatures. A thermometer placed in the water reveals that the water is still warming up even though there is no temperature change in the para-dichlorbenzene. So heat is definitely being transferred from the para-dichlorbenzene to the water. But why does the temperature of the para-dichlorbenzene remain constant during this crystallization period? Before the para-dichlorbenzene can continue to lower its temperature, it must first transition from the liquid state to the solid state. The crystallization of para-dichlorbenzene occurs at 54°C - the freezing point of the substance. At this temperature, the energy that is lost by the para-dichlorbenzene is associated with a change in the other form of internal energy - potential energy. A substance not only possesses kinetic energy due to the motion of its particles, it also possesses potential energy due to the intermolecular attractions between particles. As the para-dichlorbenzene crystallizes at 54°C, the energy being lost is reflected by decreases in the potential energy of the para-dichlorbenzene as it changes state. Once all the para-dichlorbenzene has changed to the solid state, the loss of energy is once more reflected by a decrease in the kinetic energy of the substance; its temperature decreases.

Hooke’s Law

The Simple Pendulum

Amplitude, Period, Frequency

Wave Types and Motion

Period, Frequency and Wave Speed

Vibrational Motion

Things wiggle. They do the back and forth. They vibrate; they shake; they oscillate. These phrases describe the motion of a variety of objects. They even describe the motion of matter at the atomic level. Even atoms wiggle - they do the back and forth. Wiggles, vibrations, and oscillations are an inseparable part of nature. In this chapter of The Physics Classroom Tutorial, we will make an effort to understand vibrational motion and its relationship to waves. An understanding of vibrations and waves is essential to understanding our physical world. Much of what we see and hear is only possible because of vibrations and waves. We see the world around us because of light waves. And we hear the world around us because of sound waves. If we can understand waves, then we will be able to understand the world of sight and sound.

To begin our ponderings of vibrations and waves, consider one of those crazy bobblehead dolls that you've likely seen at baseball stadiums or novelty shops. A bobblehead doll consists of an oversized replica of a person's head attached by a spring to a body and a stand. A light tap to the oversized head causes it to bobble. The head wiggles; it vibrates; it oscillates. When pushed or somehow disturbed, the head does the back and forth. The back and forth doesn't happen forever. Over time, the vibrations tend to die off and the bobblehead stops bobbing and finally assumes its usual resting position.

The bobblehead doll is a good illustration of many of the principles of vibrational motion. Think about how you would describe the back and forth motion of the oversized head of a bobblehead doll. What words would you use to describe such a motion? How does the motion of the bobblehead change over time? How does the motion of one bobblehead differ from the motion of another bobblehead? What quantities could you measure to describe the motion and so distinguish one motion from another motion? How would you explain the cause of such a motion? Why does the back and forth motion of the bobblehead finally stop? These are all questions worth pondering and answering if we are to understand vibrational motion.

What Causes Objects to Vibrate?

Like any object that undergoes vibrational motion, the bobblehead has a resting position. The resting position is the position assumed by the bobblehead when it is not vibrating. The resting position is sometimes referred to as theequilibrium position. When an object is positioned at its equilibrium position, it is in a state of equilibrium. An object which is in a state of equilibrium is experiencing a balance of forces. All the individual forces - gravity, spring, etc. - are balanced or add up to an overall net force of 0 Newtons. When a bobblehead is at the equilibrium position, the forces on the bobblehead are balanced. The bobblehead will remain in this position until somehow disturbed from its equilibrium.

If a force is applied to the bobblehead, the equilibrium will be disturbed and the bobblehead will begin vibrating. We could use the phrase forced vibration to describe the force which sets the otherwise resting bobblehead into motion. In this case, the force is a short-lived, momentary force that begins the motion. The bobblehead does its back and forth, repeating the motion over and over. Each repetition of its back and forth motion is a little less vigorous than its previous repetition. If the head sways 3 cm to the right of its equilibrium position during the first repetition, it may only sway 2.5 cm to the right of its equilibrium position during the second repetition. And it may only sway 2.0 cm to the right of its equilibrium position during the third repetition. And so on. The extent of its displacement from the equilibrium position becomes less and less over time. Because the forced vibration that initiated the motion is a single instance of a short-lived, momentary force, the vibrations ultimately cease. The bobblehead is said to experiencedamping. Damping is the tendency of a vibrating object to lose or to dissipate its energy over time. The mechanical energy of the bobbing head is lost to other objects. Without a sustained forced vibration, the back and forth motion of the bobblehead eventually ceases as energy is dissipated to other objects. A sustained input of energy would be required to keep the back and forth motion going. After all, if the vibrating object naturally loses energy, then it must continuously be put back into the system through a forced vibration in order to sustain the vibration.

The Restoring Force

A vibrating bobblehead often does the back and forth a number of times. The vibrations repeat themselves over and over. As such, the bobblehead will move back to (and past) the equilibrium position every time it returns from its maximum displacement to the right or the left (or above or below). This begs a question - and perhaps one that you have been thinking of yourself as you've pondered the topic of vibration. If the forces acting upon the bobblehead are balanced when at the equilibrium position, then why does the bobblehead sway past this position? Why doesn't the bobblehead stop the first time it returns to the equilibrium position? The answer to this question can be found in Newton's first law of motion. Like any moving object, the motion of a vibrating object can be understood in light of Newton's laws. According to Newton's law of inertia, an object which is moving will continue its motion if the forces are balanced. Put another way, forces, when balanced, do not stop moving objects. So every instant in time that the bobblehead is at the equilibrium position, the momentary balance of forces will not stop the motion. The bobblehead keeps moving. It moves past the equilibrium position towards the opposite side of its swing. As the bobblehead is displaced past its equilibrium position, then a force capable of slowing it down and stopping it exists. This force that slows the bobblehead down as it moves away from its equilibrium position is known as a restoring force. The restoring force acts upon the vibrating object to move it back to its original equilibrium position.

Vibrational motion is often contrasted with translational motion. In translational motion, an object is permanently displaced. The initial force that is imparted to the object displaces it from its resting position and sets it into motion. Yet because there is no restoring force, the object continues the motion in its original direction. When an object vibrates, it doesn't move permanently out of position. The restoring force acts to slow it down, change its direction and force it back to its original equilibrium position. An object in translational motion is permanently displaced from its original position. But an object in vibrational motion wiggles about a fixed position - its original equilibrium position. Because of the restoring force, vibrating objects do the back and forth.

Other Vibrating Systems

As you know, bobblehead dolls are not the only objects that vibrate. It might be safe to say that all objects in one way or another can be forced to vibrate to some extent. The vibrations might not be large enough to be visible. Or the amount of damping might be so strong that the object scarcely completes a full cycle of vibration. But as long as a force persists to restore the object to its original position, a displacement from its resting position will result in a vibration. Even a large massive skyscraper is known to vibrate as winds push upon its structure. While held fixed in place at its foundation (we hope), the winds force the length of the structure out of position and the skyscraper is forced into vibration.

A pendulum is a classic example of an object that is considered to vibrate. A simple pendulum consists of a relatively massive object hung by a string from a fixed support. It typically hangs vertically in its equilibrium position. When the mass is displaced from equilibrium, it begins its back and forth vibration about its fixed equilibrium position. The motion is regular and repeating. In the next part of this lesson, we will describe such a regular and repeating motion as a periodic motion. Because of the regular nature of a pendulum's motion, many clocks, such as grandfather clocks, use a pendulum as part of its timing mechanism.

An inverted pendulum is another classic example of an object that undergoes vibrational motion. An inverted pendulum is simply a pendulum which has its fixed end located below the vibrating mass. An inverted pendulum can be made by attaching a mass (such as a tennis ball) to the top end of a dowel rod and then securing the bottom end of the dowel rod to a horizontal support. This is shown in the diagram below. A gentle force exerted upon the tennis ball will cause it to vibrate about a fixed, equilibrium position. The vibrating skyscraper can be thought of as a type of inverted pendulum. Tall trees are often displaced from their usual vertical orientation by strong winds. As the winds cease, the trees will vibrate back and forth about their fixed positions. Such trees can be thought of as acting as inverted pendula. Even the tines of a tuning fork can be considered a type of inverted pendulum

Another classic example of an object that undergoes vibrational motion is a mass on a spring. The animation at the right depicts a mass suspended from a spring. The mass hangs at a resting position. If the mass is pulled down, the spring is stretched. Once the mass is released, it begins to vibrate. It does the back and forth, vibrating about a fixed position. If the spring is rotated horizontally and the mass is placed upon a supporting surface, the same back and forth motion can be observed. Pulling the mass to the right of its resting position stretches the spring. When released, the mass is pulled back to the left, heading towards its resting position. After passing by its resting position, the spring begins to compress. The compressions of the coiled spring result in a restoring force that again pushes rightward on the leftward moving mass. The cycle continues as the mass vibrates back and forth about a fixed position. The springs inside of a bed mattress, the suspension systems of some cars, and bathroom scales all operated as a mass on a spring system.

In all the vibrating systems just mentioned, damping is clearly evident. The simple pendulum doesn't vibrate forever; its energy is gradually dissipated through air resistance and loss of energy to the support. The inverted pendulum consisting of a tennis ball mounted to the top of a dowel rod does not vibrate forever. Like the simple pendulum, the energy of the tennis ball is dissipated through air resistance and vibrations of the support. Frictional forces also cause the mass on a spring to lose its energy to the surroundings. In some instances, damping is a favored feature. Car suspension systems are intended to dissipate vibrational energy, preventing drivers and passengers from having to do the back and forth as they also do the down the road.

Hopefully a lot of our original questions have been answered. But one question that has not yet been answered is the question pertaining to quantities that can be measured. How can we quantitatively describe a vibrating object? What measurements can be made of vibrating objects that would distinguish one vibrating object from another? We will ponder this question in the next part of this lesson on vibrational motion.

Properties of Periodic Motion

A vibrating object is wiggling about a fixed position. Like the mass on a spring in the animation at the right, a vibrating object is moving over the same path over the course of time. Its motion repeats itself over and over again. If it were not for damping, the vibrations would endure forever (or at least until someone catches the mass and brings it to rest). The mass on the spring not only repeats the same motion, it does so in a regular fashion. The time it takes to complete one back and forth cycle is always the same amount of time. If it takes the mass 3.2 seconds for the mass to complete the first back and forth cycle, then it will take 3.2 seconds to complete the seventh back and forth cycle. It's like clockwork. It's so predictable that you could set your watch by it. In Physics, a motion that is regular and repeating is referred to as a periodic motion. Most objects that vibrate do so in a regular and repeated fashion; their vibrations are periodic.

One obvious characteristic of the graph has to do with its shape. Many students recognize the shape of this graph from experiences in Mathematics class. The graph has the shape of a sine wave. If y = sine(x) is plotted on a graphing calculator, a graph with this same shape would be created. The vertical axis of the above graph represents the position of the mass relative to the motion detector. A position of about 0.60 m cm above the detector represents the resting position of the mass. So the mass is vibrating back and forth about this fixed resting position over the course of time. There is something sinusoidal about the vibration of a mass on a spring. And the same can be said of a pendulum vibrating about a fixed position or of a guitar string or of the air inside of a wind instrument. The position of the mass is a function of the sine of the time.

A second obvious characteristic of the graph may be its periodic nature. The motion repeats itself in a regular fashion. Time is being plotted along the horizontal axis; so any measurement taken along this axis is a measurement of the time for something to happen. A full cycle of vibration might be thought of as the movement of the mass from its resting position (A) to its maximum height (B), back down past its resting position (C) to its minimum position (D), and then back to its resting position (E). Using measurements from along the time axis, it is possible to determine the time for one complete cycle. The mass is at position A at a time of 0.0 seconds and completes its cycle when it is at position E at a time of 2.3 seconds. It takes 2.3 seconds to complete the first full cycle of vibration. Now if the motion of this mass is periodic (i.e., regular and repeating), then it should take the same time of 2.3 seconds to complete any full cycle of vibration. The same time-axis measurements can be taken for the sixth full cycle of vibration. In the sixth full cycle, the mass moves from a resting position (U) up to V, back down past W to X and finally back up to its resting position (Y) in the time interval from 11.6 seconds to 13.9 seconds. This represents a time of 2.3 seconds to complete the sixth full cycle of vibration. The two cycle times are identical. Other cycle times are indicated in the table below. By inspection of the table, one can safely conclude that the motion of the mass on a spring is regular and repeating; it is clearly periodic. The small deviation from 2.3 s in the third cyle can be accounted for by the lack of precision in the reading of the graph.

A third obvious characteristic of the graph is that damping occurs with the mass-spring system. Some energy is being dissipated over the course of time. The extent to which the mass moves above (B, F, J, N, R and V) or below (D, H, L, P, T and X) the resting position (C, E, G, I, etc.) varies over the course of time. In the first full cycle of vibration being shown, the mass moves from its resting position (A) 0.60 m above the motion detector to a high position (B) of 0.99 m cm above the motion detector. This is a total upward displacement of 0.29 m. In the sixth full cycle of vibration that is shown, the mass moves from its resting position (U) 0.60 m above the motion detector to a high position (V) 0.94 m above the motion detector. This is a total upward displacement of 0.24 m cm. The table below summarizes displacement measurements for several other cycles displayed on the graph.

Over the course of time, the mass continues to vibrate - moving away from and back towards the original resting position. However, the amount of displacement of the mass at its maximum and minimum height is decreasing from one cycle to the next. This illustrates that energy is being lost from the mass-spring system. If given enough time, the vibration of the mass will eventually cease as its energy is dissipated.

So far in this part of the lesson, we have looked at measurements of time and position of a mass on a spring. The measurements were based upon readings of a position-time graph. The data on the graph was collected by a motion detector that was capturing a history of the motion over the course of time. The key measurements that have been made are measurements of:

the time for the mass to complete a cycle, and

the maximum displacement of the mass above (or below) the resting position.

These two measurable quantities have names. We call these quantities period and amplitude.

Period and Frequency

An object that is in periodic motion - such as a mass on a spring, a pendulum or a bobblehead doll - will undergo back and forth vibrations about a fixed position in a regular and repeating fashion. The fact that the periodic motion is regular and repeating means that it can be mathematically described by a quantity known as the period. The period of the object's motion is defined as the time for the object to complete one full cycle. Being a time, the period is measured in units such as seconds, milliseconds, days or even years. The standard metric unit for period is the second.

An object in periodic motion can have a long period or a short period. For instance, a pendulum bob tied to a 1-meter length string has a period of about 2.0 seconds. For comparison sake, consider the vibrations of a piano string that plays themiddle C note (the C note of the fourth octave). Its period is approximately 0.0038 seconds (3.8 milliseconds). When comparing these two vibrating objects - the 1.0-meter length pendulum and the piano string which plays the middle C note - we would describe the piano string as vibrating relatively frequently and we would describe the pendulum as vibrating relatively infrequently. Observe that the description of the two objects uses the termsfrequently and infrequently. The terms fast and slow are not used since physics types reserve the words fast and slow to refer to an object's speed. Here in this description we are referring to the frequency, not the speed. An object can be in periodic motion and have a low frequency and a high speed. As an example, consider the periodic motion of the moon in orbit about the earth. The moon moves very fast; its orbit is highly infrequent. It moves through space with a speed of about 1000 m/s - that's fast. Yet it makes a complete cycle about the earth once every 27.3 days (a period of about 2.4x105 seconds) - that's infrequent.

The concept and quantity frequency is best understood if you attach it to the everyday English meaning of the word. Frequency is a word we often use to describe how often something occurs. You might say that you frequently check your email or you frequently talk to a friend or you frequently wash your hands when working with chemicals. Used in this context, you mean that you do these activities often. To say that you frequently check your email means that you do it several times a day - you do it often. In physics, frequency is used with the same meaning - it indicates how often a repeated event occurs. High frequency events that are periodic occur often, with little time in between each occurrence - like the back and forth vibrations of the tines of a tuning fork. The vibrations are so frequent that they can't be seen with the naked eye. A 256-Hz tuning fork has tines that make 256 complete back and forth vibrations each second. At this frequency, it only takes the tines about 0.00391 seconds to complete one cycle. A 512-Hz tuning fork has an even higher frequency. Its vibrations occur more frequently; the time for a full cycle to be completed is 0.00195 seconds. In comparing these two tuning forks, it is obvious that the tuning fork with the highest frequency has the lowest period. The two quantities frequency and period are inversely related to each other. In fact, they are mathematical reciprocals of each other. The frequency is the reciprocal of the period and the period is the reciprocal of the frequency.

This reciprocal relationship is easy to understand. After all, the two quantities are conceptual reciprocals (a phrase I made up). Consider their definitions as restated below:

period = the time for one full cycle to complete itself; i.e., seconds/cycle

frequency = the number of cycles that are completed per time; i.e., cycles/second

Amplitude of Vibration: The final measurable quantity that describes a vibrating object is the amplitude. The amplitude is defined as the maximum displacement of an object from its resting position. The resting position is that position assumed by the object when not vibrating. Once vibrating, the object oscillates about this fixed position. If the object is a mass on a spring (such as the discussion earlier on this page), then it might be displaced a maximum distance of 35 cm below the resting position and 35 cm above the resting position. In this case, the amplitude of motion is 35 cm.

Over the course of time, the amplitude of a vibrating object tends to become less and less. The amplitude of motion is a reflection of the quantity of energy possessed by the vibrating object. An object vibrating with a relatively large amplitude has a relatively large amount of energy. Over time, some of this energy is lost due to damping. As the energy is lost, the amplitude decreases. If given enough time, the amplitude decreases to 0 as the object finally stops vibrating. At this point in time, it has lost all its energy.

Fluids Density and Buoyant Force Archimedes Principle Fluid Pressure

Fluids

A fluid is a substance that cannot maintain its own shape but takes the shape of its container. Fluid laws assume idealized fluids that cannot be compressed.

Density and pressure

The density (ρ) of a substance of uniform composition is its mass per unit volume: ρ = m/ V. In the SI system, density is measured in units of kilograms per cubic meter. Imagine an upright cylindrical beaker filled with a fluid. The fluid exerts a force on the bottom of the container due to its weight. Pressure is defined as the force per unit area: P

F/ A, or in terms of magnitude, P

mg/A, where mg is the weight of the fluid. The SI unit of pressure is N/m2, called a pascal. The pressure at the bottom of a fluid can be expressed in terms of the density (ρ) and height (h) of the fluid: or P = ρhg. The pressure at any point in a fluid acts equally in all directions. This concept is sometimes called the basic law of fluid pressure.

Archimedes' principle

Water commonly provides partial support for any object placed in it. The upward force on an object placed in a fluid is called the buoyant force. According to Archimedes' principle, the magnitude of a buoyant force on a completely or partially submerged object always equals the weight of the fluid displaced by the object. Archimedes' principle can be verified by a nonmathematical argument. Consider the cubic volume of water in the container of water shown in Figure 2 . This volume is in equilibrium with the forces acting on it, which are the weight and the buoyant force; therefore, the downward force of the weight ( W) must be balanced by the upward buoyant force ( B), which is provided by the rest of the water in the container.

If a solid floats partially submerged in a liquid, the volume of liquid displaced is less than the volume of the solid. Comparing the density of the solid and the liquid in which it floats leads to an interesting result. The formulas for density are Ds = ms/ Vs and Dl = ml/ Vl, where D is the density, V is the volume, m is the mass, and the subscripts s and l refer to quantities associated with the solid and the liquid respectively. Solving for the masses leads to ms = DsVs and ml = DlVl. According to Archimedes' principle, the weights of the solid and the displaced liquid are equal. Because the weights are simply mass times a constant (g), the masses must be equal also; therefore, DsVs= DlVl or Ds/Dl = VlVl. Now, V = Ah, where A is the cross-sectional area and h is the height. For a solid floating in liquid, Al=As and hl is the height of the solid that is submerged, hsub. With these substitutions, the above relationship becomes Ds/Dl = hsub/hs; therefore, the fractional part of the solid that is submerged is equal to the ratio of the density of the solid to the density of the surrounding liquid in which it floats. For example, about 90 percent of an iceberg is beneath the surface of sea water because the density of ice is about nine-tenths that of sea water.

Pascal's principle

Pascal's principle may be stated thus: The pressure applied at one point in an enclosed fluid under equilibrium conditions is transmitted equally to all parts of the fluid. This rule is utilized in hydraulic systems.

Let the subscripts a and b denote the quantities at each piston. The pressures are equal; therefore, Pa = Pb. Substitute the expression for pressure in terms of force and area to obtain fa/ Aa = ( Fb/ Ab). Substitute π r2 for the area of a circle, simplify, and solve for Fb: Fb=( Fa)( rb2/ ra2). Because the force exerted at point a is multiplied by the square of the ratio of the radii and rb > ra, a modest force on the small piston a can lift a relatively larger weight on piston b.

Bernoulli's equation Imagine a fluid flowing through a section of pipe with one end having a smaller cross-sectional area than the pipe at the other end. The flow of liquids is very complex; therefore, this discussion will assume conditions of the smooth flow of an incompressible fluid through walls with no drag. The velocity of the fluid in the constricted end must be greater than the velocity at the larger end if steady flow is maintained; that is, the volume passing per time is the same at all points. Swiftly moving fluids exert less pressure than slowly moving fluids. Bernoulli's equation applies conservation of energy to formalize this observation: P + (1/2) ρ v2 + ρ gh = a constant. The equation states that the sum of the pressure (P), the kinetic energy per unit volume, and the potential energy per unit volume have the same value throughout the pipe.

Rotational motion
Torque Analysis and addition Simple Machines Torque

Torque is a measure of how much a force acting on an object causes that object to rotate. The object rotates about an axis, which we will call the pivot point, and will label 'O'. We will call the force 'F'. The distance from the pivot point to the point where the force acts is called the moment arm, and is denoted by 'r'. Note that this distance, 'r', is also a vector, and points from the axis of rotation to the point where the force acts. (Refer to Figure 1 for a pictoral representation of these definitions.)

Figure 1 Definitions

Torque is defined as
= r x F = r F sin().
In other words, torque is the cross product between the distance vector (the distance from the pivot point to the point where force is applied) and the force vector, 'a' being the angle between r and F. Using the right hand rule, we can find the direction of the torque vector. If we put our fingers in the direction of r, and curl them to the direction of F, then the thumb points in the direction of the torque vector.
Imagine pushing a door to open it. The force of your push (F) causes the door to rotate about its hinges (the pivot point, O). How hard you need to push depends on the distance you are from the hinges (r) (and several other things, but let's ignore them now). The closer you are to the hinges (i.e. the smaller r is), the harder it is to push. This is what happens when you try to push open a door on the wrong side. The torque you created on the door is smaller than it would have been had you pushed the correct side (away from its hinges). Note that the force applied, F, and the moment arm, r, are independent of the object. Furthermore, a force applied at the pivot point will cause no torque since the moment arm would be zero (r = 0).

Another way of expressing the above equation is that torque is the product of the magnitude of the force and the perpendicular distance from the force to the axis of rotation (i.e. the pivot point). Let the force acting on an object be broken up into its tangential (Ftan) and radial (Frad) components (see Figure 2). (Note that the tangential component isperpendicular to the moment arm, while the radial component is parallel to the moment arm.) The radial component of the force has no contribution to the torque because it passes through the pivot point. So, it is only the tangential component of the force which affects torque (since it is perpendicular to the line between the point of action of the force and the pivot point).

Figure 2 Tangential and radial components of force F

There may be more than one force acting on an object, and each of these forces may act on different point on the object. Then, each force will cause a torque. The net torque is the sum of the individual torques.
Rotational Equilibrium is analogous to translational equilibrium, where the sum of the forces are equal to zero. In rotational equilibrium, the sum of the torques is equal to zero. In other words, there is no net torque on the object.

Note that the SI units of torque is a Newton-metre, which is also a way of expressing a Joule (the unit for energy). However, torque is not energy. So, to avoid confusion, we will use the units N.m, and not J. The distinction arises because energy is a scalar quanitity, whereas torque is a vector.

Centripetal Acceleration and Force Gravitational Force and its applications Speed and Velocity

The same concepts and principles used to describe and explain the motion of an object can be used to describe and explain the parabolic motion of a projectile. In this unit, we will see that these same concepts and principles can also be used to describe and explain the motion of objects that either move in circles or can be approximated to be moving in circles. Kinematic concepts and motion principles will be applied to the motion of objects in circles and then extended to analyze the motion of such objects as roller coaster cars, a football player making a circular turn, and a planet orbiting the sun. We will see that the beauty and power of physics lies in the fact that a few simple concepts and principles can be used to explain the mechanics of the entire universe. Lesson 1 of this study will begin with the development of kinematic and dynamic ideas that can be used to describe and explain the motion of objects in circles. Uniform circular motion is the motion of an object in a circle with a constant or uniform speed.

Calculation of the Average Speed

Uniform circular motion - circular motion at a constant speed - is one of many forms of circular motion. An object moving in uniform circular motion would cover the same linear distance in each second of time. When moving in a circle, an object traverses a distance around the perimeter of the circle. So if your car were to move in a circle with a constant speed of 5 m/s, then the car would travel 5 meters along the perimeter of the circle in each second of time. The distance of one complete cycle around the perimeter of a circle is known as the circumference. With a uniform speed of 5 m/s, a car could make a complete cycle around a circle that had a circumference of 5 meters. At this uniform speed of 5 m/s, each cycle around the 5-m circumference circle would require 1 second. At 5 m/s, a circle with a circumference of 20 meters could be made in 4 seconds; and at this uniform speed, every cycle around the 20-m circumference of the circle would take the same time period of 4 seconds. This relationship between the circumference of a circle, the time to complete one cycle around the circle, and the speed of the object is merely an extension of the average speed equation. The circumference of any circle can be computed using from the radius according to the equation.

Combining these two equations above will lead to a new equation relating the speed of an object moving in uniform circular motion to the radius of the circle and the time to make one cycle around the circle (period).

external image u6l1e1.gif

where R represents the radius of the circle and T represents the period. This equation, like all equations, can be used as an algebraic recipe for problem solving. It also can be used to guide our thinking about the variables in the equation relate to each other. For instance, the equation suggests that for objects moving around circles of different radius in the same period, the object traversing the circle of larger radius must be traveling with the greatest speed. In fact, the average speed and the radius of the circle are directly proportional. A twofold increase in radius corresponds to a twofold increase in speed; a threefold increase in radius corresponds to a three--fold increase in speed; and so on. To illustrate, consider a strand of four LED lights positioned at various locations along the strand. The strand is held at one end and spun rapidly in a circle. Each LED light traverses a circle of different radius. Yet since they are connected to the same wire, their period of rotation is the same. Subsequently, the LEDs that are further from the center of the circle are traveling faster in order to sweep out the circumference of the larger circle in the same amount of time. If the room lights are turned off, the LEDs created an arc that could be perceived to be longer for those LEDs that were traveling faster - the LEDs with the greatest radius. This is illustrated in the diagram at the right.

The Direction of the Velocity Vector

Speed is a scalar quantity and velocity is a vector quantity. Velocity, being a vector, has both a magnitude and a direction. The magnitude of the velocity vector is the instantaneous speed of the object. The direction of the velocity vector is directed in the same direction that the object moves. Since an object is moving in a circle, its direction is continuously changing. At one moment, the object is moving northward such that the velocity vector is directed northward. One quarter of a cycle later, the object would be moving eastward such that the velocity vector is directed eastward. As the object rounds the circle, the direction of the velocity vector is different than it was the instant before. So while the magnitude of the velocity vector may be constant, the direction of the velocity vector is changing. The best word that can be used to describe the direction of the velocity vector is the word tangential. The direction of the velocity vector at any instant is in the direction of a tangent line drawn to the circle at the object's location. (A tangent line is a line that touches a circle at one point but does not intersect it.) The diagram at the right shows the direction of the velocity vector at four different points for an object moving in a clockwise direction around a circle. While the actual direction of the object (and thus, of the velocity vector) is changing, its direction is always tangent to the circle. To summarize, an object moving in uniform circular motion is moving around the perimeter of the circle with a constant speed. While the speed of the object is constant, its velocity is changing. Velocity, being a vector, has a constant magnitude but a changing direction. The direction is always directed tangent to the circle and as the object turns the circle, the tangent line is always pointing in a new direction.

Acceleration An object moving in uniform circular motion is moving in a circle with a uniform or constant speed. The velocity vector is constant in magnitude but changing in direction. Because the speed is constant for such a motion, many students have the misconception that there is no acceleration. "After all," they might say, "if I were driving a car in a circle at a constant speed of 20 mi/hr, then the speed is neither decreasing nor increasing; therefore there must not be an acceleration." At the center of this common student misconception is the wrong belief that acceleration has to do with speed and not with velocity. But the fact is that an accelerating object is an object that is changing its velocity. And since velocity is a vector that has both magnitude and direction, a change in either the magnitude or the direction constitutes a change in the velocity. For this reason, it can be safely concluded that an object moving in a circle at constant speed is indeed accelerating. It is accelerating because the direction of the velocity vector is changing. To understand this at a deeper level, we will have to combine the definition of acceleration with a review of some basic vector principles. Acceleration as a quantity was defined as the rate at which the velocity of an object changes. As such, it is calculated using the following equation:

external image def-acceleration.gif

where vi represents the initial velocity and vf represents the final velocity after some time of t. The numerator of the equation is found by subtracting one vector (vi) from a second vector (vf). But the addition and subtraction of vectors from each other is done in a manner much different than the addition and subtraction of scalar quantities. Consider the case of an object moving in a circle about point C as shown in the diagram below. In a time of t seconds, the object has moved from point A to point B. In this time, the velocity has changed from vi to vf. The process of subtracting vi from vf is shown in the vector diagram; this process yields the change in velocity.

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The Centripetal Force Requirement
An object moving in a circle is experiencing an acceleration. Even if moving around the perimeter of the circle with a constant speed, there is still a change in velocity and subsequently an acceleration. This acceleration is directed towards the center of the circle. And in accord with Newton's second law of motion , an object which experiences an acceleration must also be experiencing a net force. The direction of the net force is in the same direction as the acceleration. So for an object moving in a circle, there must be an inward force acting upon it in order to cause its inward acceleration. This is sometimes referred to as the centripetal force requirement . The word centripetal (not to be confused with the F-wordcentrifugal) means center seeking. For object's moving in circular motion, there is a net force acting towards the center which causes the object to seek the center. To understand the importance of a centripetal force, it is important to have a sturdy understanding of the Newton's first law of motion - the law of inertia. The law of inertia states that ...

|| ... objects in motion tend to stay in motion with the same speed and the same direction unless acted upon by an unbalanced force. ||

According to Newton's first law of motion, it is the natural tendency of all moving objects to continue in motion in the same direction that they are moving ... unless some form of unbalanced force acts upon the object to deviate its motion from its straight-line path. Moving objects will tend to naturally travel in straight lines; an unbalanced force is only required to cause it to turn. Thus, the presence of an unbalanced force is required for objects to move in circles.

The Centripetal Force and Direction Change

Any object moving in a circle (or along a circular path) experiences a centripetal force. That is, there is some physical force pushing or pulling the object towards the center of the circle. This is the centripetal force requirement. The wordcentripetal is merely an adjective used to describe the direction of the force. We are not introducing a new type of force but rather describing the direction of the net force acting upon the object that moves in the circle. Whatever the object, if it moves in a circle, there is some force acting upon it to cause it to deviate from its straight-line path, accelerate inwards and move along a circular path. Three such examples of centripetal force are shown below.

external image u6l2a2.gif

external image u6l1c3.gif

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As a car makes a turn, the force of friction acting upon the turned wheels of the car provides centripetal force required for circular motion.

As a bucket of water is tied to a string and spun in a circle, the tension force acting upon the bucket provides the centripetal force required for circular motion.

As the moon orbits the Earth, the force of gravity acting upon the moon provides the centripetal force required for circular motion.

Newton's Law of Universal Gravitation

Isaac Newton compared the acceleration of the moon to the acceleration of objects on earth. Believing that gravitational forces were responsible for each, Newton was able to draw an important conclusion about the dependence of gravity upon distance. This comparison led him to conclude that the force of gravitational attraction between the Earth and other objects is inversely proportional to the distance separating the earth's center from the object's center. But distance is not the only variable affecting the magnitude of a gravitational force. Consider Newton's famous equation

Fnet = m • a

Newton knew that the force that caused the apple's acceleration (gravity) must be dependent upon the mass of the apple. And since the force acting to cause the apple's downward acceleration also causes the earth's upward acceleration (Newton's third law), that force must also depend upon the mass of the earth. So for Newton, the force of gravity acting between the earth and any other object is directly proportional to the mass of the earth, directly proportional to the mass of the object, and inversely proportional to the square of the distance that separates the centers of the earth and the object.

external image UniGra.jpg

The constant of proportionality (G) in the above equation is known as the universal gravitation constant. The precise value of G was determined experimentally by Henry Cavendish in the century after Newton's death. The value of G is found to be G = 6.673 x 10-11 N m2/kg2

The units on G may seem rather odd; nonetheless they are sensible. When the units on G are substituted into the equation above and multiplied by m1• m2units and divided by d2 units, the result will be Newtons - the unit of force.

Knowing the value of G allows us to calculate the force of gravitational attraction between any two objects of known mass and known separation distance.

Forces in a Collision
Collisions
Types of Collisions
Perfectly Inelastic Collisions
Kinetic Energy in Inelastic Collisions Elastic Collisions

Inelastic Collision

Collisions between objects are governed by laws of momentum and energy. When a collision occurs in an isolated system, the total momentum of the system of objects is conserved. Provided that there are no net external forces acting upon the objects, the momentum of all objects before the collision equals the momentum of all objects after the collision. If there are only two objects involved in the collision, then the momentum change of the individual objects are equal in magnitude and opposite in direction.

Certain collisions are referred to as elastic collisions. Elastic collisions are collisions in which both momentum and kinetic energy are conserved. The total system kinetic energy before the collision equals the total system kinetic energy after the collision. If total kinetic energy is not conserved, then the collision is referred to as an inelastic collision.

The animation below portrays the inelastic collision between a 1000-kg car and a 3000-kg truck. The before- and after-collision velocities and momentum are shown in the data tables.

In the collision between the truck and the car, total system momentum is conserved. Before the collision, the momentum of the car is +20000 kg*m/s and the momentum of the truck is -60000 kg*m/s; the total system momentum is -40000 kg*m/s. After the collision, the momentum of the car is -10000 kg*m/s and the momentum of the truck is -30 000 kg*m/s; the total system momentum is -40000 kg*m/s. The total system momentum is conserved. The momentum change of the car (-30000 kg*m/s) is equal in magnitude and opposite in direction to the momentum change of the truck (+30000 kg*m/s) .

An analysis of the kinetic energy of the two objects reveals that the total system kinetic energy before the collision is 800000 Joules (200000 J for the car plus 600000 J for the truck). After the collision, the total system kinetic energy is 200000 Joules (50000 J for the car and 150000 J for the truck). The total kinetic energy before the collision is not equal to the total kinetic energy after the collision. A large portion of the kinetic energy is converted to other forms of energy such as sound energy and thermal energy. A collision in which total system kinetic energy is not conserved is known as an inelastic collision.

Elastic Collision

Collisions between objects are governed by laws of momentum and energy. When a collision occurs in an isolated system, the total momentum of the system of objects is conserved. Provided that there are no net external forces acting upon the objects, the momentum of all objects before the collision equals the momentum of all objects after the collision. If there are only two objects involved in the collision, then the momentum change of the individual objects are equal in magnitude and opposite in direction.

Certain collisions are referred to as elastic collisions. Elastic collisions are collisions in which both momentum and kinetic energy are conserved. The total system kinetic energy before the collision equals the total system kinetic energy after the collision. If total kinetic energy is not conserved, then the collision is referred to as an inelastic collision.

The animation below portrays the elastic collision between a 1000-kg car and a 3000-kg truck. The before- and after-collision velocities and momentum are shown in the data tables.

In the collision between the truck and the car, total system momentum is conserved. Before the collision, the momentum of the car is +20000 kg*m/s and the momentum of the truck is -60000 kg*m/s; the total system momentum is -40000 kg*m/s. After the collision, the momentum of the car is -40000 kg*m/s and the momentum of the truck is 0 kg*m/s; the total system momentum is -40000 kg*m/s. The total system momentum is conserved. The momentum change of the car (-40000 kg*m/s) is equal in magnitude and opposite in direction to the momentum change of the truck (40000 kg*m/s) .

An analysis of the kinetic energy of the two objects reveals that the total system kinetic energy before the collision is 800000 Joules (200000 J for the car plus 600000 J for the truck). After the collision, the total system kinetic energy is 800000 Joules (800000 J for the car and 0 J for the truck). The total kinetic energy before the collision is equal to the total kinetic energy after the collision. A collision such as this in which total system kinetic energy is conserved is known as an elastic collision.

Momentum
Impulse
Impulse-Momentum Theorem
Law of Conservation of Momentum

Momentum

The sports announcer says, "Going into the all-star break, the Chicago White Sox have the momentum." The headlines declare "Chicago Bulls Gaining Momentum." The coach pumps up his team at half-time, saying "You have the momentum; the critical need is that you use thatmomentum and bury them in this third quarter."

Momentum is a commonly used term in sports. A team that has the momentum is on the move and is going to take some effort to stop. A team that has a lot of momentum is really on the move and is going to be hard to stop. Momentum is a physics term; it refers to the quantity of motion that an object has. A sports team that is on the movehas the momentum. If an object is in motion (on the move) then it has momentum.

Momentum can be defined as "mass in motion." All objects have mass; so if an object is moving, then it has momentum - it has its mass in motion. The amount of momentum that an object has is dependent upon two variables: how much stuff is moving and how fast the stuff is moving. Momentum depends upon the variables mass and velocity.In terms of an equation, the momentum of an object is equal to the mass of the object times the velocity of the object.

Momentum = mass • velocity

In physics, the symbol for the quantity momentum is the lower case "p". Thus, the above equation can be rewritten as

p = m • v

where m is the mass and v is the velocity. The equation illustrates that momentum is directly proportional to an object's mass and directly proportional to the object's velocity.

The units for momentum would be mass units times velocity units. The standard metric unit of momentum is the kg•m/s. While the kg•m/s is the standard metric unit of momentum, there are a variety of other units that are acceptable (though not conventional) units of momentum. Examples include kg•mi/hr, kg•km/hr, and g•cm/s. In each of these examples, a mass unit is multiplied by a velocity unit to provide a momentum unit. This is consistent with the equation for momentum.

Momentum is a vector quantity. A vector quantity is a quantity that is fully described by both magnitude and direction. To fully describe the momentum of a 5-kg bowling ball moving westward at 2 m/s, youmust include information about both the magnitude and the direction of the bowling ball. It is not enough to say that the ball has 10 kg•m/s of momentum; the momentum of the ball is not fully described until information about its direction is given. The direction of the momentum vector is the same as the direction of the velocity of the ball. In a previous unit, it was said that the direction of the velocity vector is the same as the direction that an object is moving. If the bowling ball is moving westward, then its momentum can be fully described by saying that it is 10 kg•m/s, westward. As a vector quantity, the momentum of an object is fully described by both magnitude and direction.

From the definition of momentum, it becomes obvious that an object has a large momentum if either its mass or its velocity is large. Both variables are of equal importance in determining the momentum of an object. Consider a Mack truck and a roller skate moving down the street at the same speed. The considerably greater mass of the Mack truck gives it a considerably greater momentum. Yet if the Mack truck were at rest, then the momentum of the least massive roller skate would be the greatest. The momentum of any object that is at rest is 0. Objects at rest do not have momentum - they do not have any "mass in motion." Both variables - mass and velocity - are important in comparing the momentum of two objects.

The momentum equation can help us to think about how a change in one of the two variables might affect the momentum of an object. Consider a 0.5-kg physics cart loaded with one 0.5-kg brick and moving with a speed of 2.0 m/s. The total mass of loaded cart is 1.0 kg and its momentum is 2.0 kg•m/s. If the cart was instead loaded with three 0.5-kg bricks, then the total mass of the loaded cart would be 2.0 kg and its momentum would be 4.0 kg•m/s. A doubling of the mass results in a doubling of the momentum.

Similarly, if the 2.0-kg cart had a velocity of 8.0 m/s (instead of 2.0 m/s), then the cart would have a momentum of 16.0 kg•m/s (instead of 4.0 kg•m/s). A quadrupling in velocity results in a quadrupling of the momentum. These two examples illustrate how the equation p = m•v serves as a "guide to thinking" and not merely a "plug-and-chug recipe for algebraic problem-solving." Momentum and Impulse Connection

When a sports announcer says that a team has the momentum they mean that the team is really on the move and is going to be hard to stop. The termmomentum is a physics concept. Any object with momentum is going to be hard to stop. To stop such an object, it is necessary to apply a force against its motion for a given period of time. The more momentum that an object has, the harder that it is to stop. Thus, it would require a greater amount of force or a longer amount of time or both to bring such an object to a halt. As the force acts upon the object for a given amount of time, the object's velocity is changed; and hence, the object's momentum is changed.

The concepts in the above paragraph should not seem like abstract information to you. You have observed this a number of times if you have watched the sport of football. In football, the defensive players apply a force for a given amount of time to stop the momentum of the offensive player who has the ball. You have also experienced this a multitude of times while driving. As you bring your car to a halt when approaching a stop sign or stoplight, the brakes serve to apply a force to the car for a given amount of time to change the car's momentum. An object with momentum can be stopped if a force is applied against it for a given amount of time.

A force acting for a given amount of time will change an object's momentum. Put another way, an unbalanced force always accelerates an object - either speeding it up or slowing it down. If the force acts opposite the object's motion, it slows the object down. If a force acts in the same direction as the object's motion, then the force speeds the object up. Either way, a force will change the velocity of an object. And if the velocity of the object is changed, then the momentum of the object is changed.

These concepts are merely an outgrowth of Newton's second law as discussed in an earlier unit. Newton's second law (Fnet = m • a) stated that the acceleration of an object is directly proportional to the net force acting upon the object and inversely proportional to the mass of the object. When combined with the definition of acceleration (a = change in velocity / time), the following equalities result.

If both sides of the above equation are multiplied by the quantity t, a new equation results.

This equation represents one of two primary principles to be used in the analysis of collisions during this unit. To truly understand the equation, it is important to understand its meaning in words. In words, it could be said that the force times the time equals the mass times the change in velocity. In physics, the quantity Force • time is known asimpulse. And since the quantity m•v is the momentum, the quantity m•Δv must be the change in momentum. The equation really says that the

Impulse = Change in momentum

One focus of this unit is to understand the physics of collisions. The physics of collisions are governed by the laws of momentum; and the first law that we discuss in this unit is expressed in the above equation. The equation is known as the impulse-momentum change equation. The law can be expressed this way:

In a collision, an object experiences a force for a specific amount of time that results in a change in momentum. The result of the force acting for the given amount of time is that the object's mass either speeds up or slows down (or changes direction). The impulse experienced by the object equals the change in momentum of the object. In equation form, F • t = m • Δ v.

The Law of Action-Reaction (Revisited) A collision is an interaction between two objects that have made contact (usually) with each other. As in any interaction, a collision results in a force being applied to the two colliding objects. Newton's laws of motion govern such collisions. Newton's third law of motion was introduced and discussed. It was said that...

... in every interaction, there is a pair of forces acting on the two interacting objects. The size of the force on the first object equals the size of the force on the second object. The direction of the force on the first object isopposite to the direction of the force on the second object. Forces always come in pairs - equal and opposite action-reaction force pairs.

Newton's third law of motion is naturally applied to collisions between two objects. In a collision between two objects, both objects experience forces that are equal in magnitude and opposite in direction. Such forces often cause one object to speed up (gain momentum) and the other object to slow down (lose momentum). According to Newton's third law, the forces on the two objects are equal in magnitude. While the forces are equal in magnitude and opposite in direction, the accelerations of the objects are not necessarily equal in magnitude. In accord with Newton's second law of motion, the acceleration of an object is dependent upon both force and mass. Thus, if the colliding objects have unequal mass, they will have unequal accelerations as a result of the contact force that results during the collision.

Consider the collision between the club head and the golf ball in the sport of golf. When the club head of a moving golf club collides with a golf ball at rest upon a tee, the force experienced by the club head is equal to the force experienced by the golf ball. Most observers of this collision have difficulty with this concept because they perceive the high speed given to the ball as the result of the collision. They are not observing unequal forces upon the ball and club head, but rather unequal accelerations. Both club head and ball experience equal forces, yet the ball experiences a greater acceleration due to its smaller mass. In a collision, there is a force on both objects that causes an acceleration of both objects. The forces are equal in magnitude and opposite in direction, yet the least massive object receives the greatest acceleration.

Consider the collision between a moving seven ball and an eight ball that is at rest in the sport of table pool. When the seven ball collides with the eight ball, each ball experiences an equal force directed in opposite directions. The rightward moving seven ball experiences a leftward force that causes it to slow down; the eight ball experiences a rightward force that causes it to speed up. Since the two balls have equal masses, they will also experience equal accelerations. In a collision, there is a force on both objects that causes an acceleration of both objects; the forces are equal in magnitude and opposite in direction. For collisions between equal-mass objects, each object experiences the same acceleration.

Consider the interaction between a male and female figure skater in pair figure skating. A woman (m = 45 kg) is kneeling on the shoulders of a man (m = 70 kg); the pair is moving along the ice at 1.5 m/s. The man gracefully tosses the woman forward through the air and onto the ice. The woman receives the forward force and the man receives a backward force. The force on the man is equal in magnitude and opposite in direction to the force on the woman. Yet the acceleration of the woman is greater than the acceleration of the man due to the smaller mass of the woman.

Many observers of this interaction have difficulty believing that the man experienced a backward force. "After all," they might argue, "the man did not move backward." Such observers are presuming that forces cause motion. In their minds, a backward force on the male skater would cause a backward motion. This is a common misconception that has been addressed elsewhere in The Physics Classroom. Forces cause acceleration, not motion. The male figure skater experiences a backwards force that causes his backwards acceleration. The male skater slows down while the woman skater speeds up. In every interaction (with no exception), there are forces acting upon the two interacting objects that are equal in magnitude and opposite in direction.

Collisions are governed by Newton's laws. The law of action-reaction (Newton's third law) explains the nature of the forces between the two interacting objects. According to the law, the force exerted by object 1 upon object 2 is equal in magnitude and opposite in direction to the force exerted by object 2 upon object 1.

WorkEnergyWork- Kinetic Energy TheoremPotential EnergyConservation of EnergyPower

Definition and Mathematics of Work When a force acts upon an object to cause a displacement of the object, it is said that work was done upon the object. There are three key ingredients to work - force, displacement, and cause. In order for a force to qualify as having done work on an object, there must be a displacement and the force must cause the displacement. There are several good examples of work that can be observed in everyday life - a horse pulling a plow through the field, a father pushing a grocery cart down the aisle of a grocery store, a freshman lifting a backpack full of books upon her shoulder, a weightlifter lifting a barbell above his head, an Olympian launching the shot-put, etc. In each case described here there is a force exerted upon an object to cause that object to be displaced.

Mathematically, work can be expressed by the following equation.

where F is the force, d is the displacement, and the angle (theta) is defined as the angle between the force and the displacement vector. Perhaps the most difficult aspect of the above equation is the angle "theta." The angle is not justany 'ole angle, but rather a very specific angle. The angle measure is defined as the angle between the force and the displacement.

The equation for work lists three variables - each variable is associated with one of the three key words mentioned in the definition of work (force, displacement, and cause). The angle theta in the equation is associated with the amount of force that causes a displacement. When a force is exerted on an object at an angle to the horizontal, only a part of the force contributes to (or causes) a horizontal displacement. Let's consider the force of a chain pulling upwards and rightwards upon Fido in order to drag Fido to the right. It is only the horizontal component of the tension force in the chain that causes Fido to be displaced to the right. The horizontal component is found by multiplying the force F by the cosine of the angle between F and d. In this sense, the cosine theta in the work equation relates to the cause factor - it selects the portion of the force that actually causes a displacement.

Units of Work Whenever a new quantity is introduced in physics, the standard metric units associated with that quantity are discussed. In the case of work (and also energy), the standard metric unit is the Joule (abbreviated J). One Joule is equivalent to one Newton of force causing a displacement of one meter. In other words,

The Joule is the unit of work. 1 Joule = 1 Newton * 1 meter 1 J = 1 N * m

In summary, work is done when a force acts upon an object to cause a displacement. Three quantities must be known in order to calculate the amount of work. Those three quantities are force, displacement and the angle between the force and the displacement.

Kinetic Energy Kinetic energy is the energy of motion. An object that has motion - whether it is vertical or horizontal motion - has kinetic energy. There are many forms of kinetic energy - vibrational (the energy due to vibrational motion), rotational (the energy due to rotational motion), and translational (the energy due to motion from one location to another). To keep matters simple, we will focus upon translational kinetic energy. The amount of translational kinetic energy (from here on, the phrase kinetic energy will refer to translational kinetic energy) that an object has depends upon two variables: the mass (m) of the object and the speed (v) of the object. The following equation is used to represent the kinetic energy (KE) of an object.

where m = mass of object v = speed of object

This equation reveals that the kinetic energy of an object is directly proportional to the square of its speed. That means that for a twofold increase in speed, the kinetic energy will increase by a factor of four. For a threefold increase in speed, the kinetic energy will increase by a factor of nine. And for a fourfold increase in speed, the kinetic energy will increase by a factor of sixteen. The kinetic energy is dependent upon the square of the speed. As it is often said, an equation is not merely a recipe for algebraic problem solving, but also a guide to thinking about the relationship between quantities.

Kinetic energy is a scalar quantity; it does not have a direction. Unlike velocity, acceleration, force, and momentum, the kinetic energy of an object is completely described by magnitude alone. Like work and potential energy, the standard metric unit of measurement for kinetic energy is the Joule. As might be implied by the above equation, 1 Joule is equivalent to 1 kg*(m/s)^2. Potential Energy An object can store energy as the result of its position. For example, the heavy ball of a demolition machine is storing energy when it is held at an elevated position. This stored energy of position is referred to as potential energy. Similarly, a drawn bow is able to store energy as the result of its position. When assuming its usual position (i.e., when not drawn), there is no energy stored in the bow. Yet when its position is altered from its usual equilibrium position, the bow is able to store energy by virtue of its position. This stored energy of position is referred to as potential energy. Potential energy is the stored energy of position possessed by an object. Gravitational Potential Energy Gravitational potential energy is the energy stored in an object as the result of its vertical position or height. The energy is stored as the result of the gravitational attraction of the Earth for the object. The gravitational potential energy of the massive ball of a demolition machine is dependent on two variables - the mass of the ball and the height to which it is raised. There is a direct relation between gravitational potential energy and the mass of an object. More massive objects have greater gravitational potential energy. There is also a direct relation between gravitational potential energy and the height of an object. The higher that an object is elevated, the greater the gravitational potential energy. These relationships are expressed by the following equation: In the above equation, m represents the mass of the object, h represents the height of the object and g represents the gravitational field strength (9.8 N/kg on Earth) - sometimes referred to as the acceleration of gravity. Since the gravitational potential energy of an object is directly proportional to its height above the zero position, a doubling of the height will result in a doubling of the gravitational potential energy. A tripling of the height will result in a tripling of the gravitational potential energy.

Elastic Potential Energy The second form of potential energy that we will discuss is elastic potential energy. Elastic potential energy is the energy stored in elastic materials as the result of their stretching or compressing. Elastic potential energy can be stored in rubber bands, bungee chords, trampolines, springs, an arrow drawn into a bow, etc. The amount of elastic potential energy stored in such a device is related to the amount of stretch of the device - the more stretch, the more stored energy.

Springs are a special instance of a device that can store elastic potential energy due to either compression or stretching. A force is required to compress a spring; the more compression there is, the more force that is required to compress it further. For certain springs, the amount of force is directly proportional to the amount of stretch or compression (x); the constant of proportionality is known as the spring constant (k).

Such springs are said to follow Hooke's Law. If a spring is not stretched or compressed, then there is no elastic potential energy stored in it. The spring is said to be at its equilibrium position. The equilibrium position is the position that the spring naturally assumes when there is no force applied to it. In terms of potential energy, the equilibrium position could be called the zero-potential energy position. There is a special equation for springs that relates the amount of elastic potential energy to the amount of stretch (or compression) and the spring constant. The equation is To summarize, potential energy is the energy that is stored in an object due to its position relative to some zero position. An object possesses gravitational potential energy if it is positioned at a height above (or below) the zero height. An object possesses elastic potential energy if it is at a position on an elastic medium other than the equilibrium position. Mechanical Energy It was said that work is done upon an object whenever a force acts upon it to cause it to be displaced. Work involves a force acting upon an object to cause a displacement. In all instances in which work is done, there is an object that supplies the force in order to do the work. If a World Civilization book is lifted to the top shelf of a student locker, then the student supplies the force to do the work on the book. If a plow is displaced across a field, then some form of farm equipment (usually a tractor or a horse) supplies the force to do the work on the plow. If a pitcher winds up and accelerates a baseball towards home plate, then the pitcher supplies the force to do the work on the baseball. If a roller coaster car is displaced from ground level to the top of the first drop of a roller coaster ride, then a chain driven by a motor supplies the force to do the work on the car. If a barbell is displaced from ground level to a height above a weightlifter's head, then the weightlifter is supplying a force to do work on the barbell. In all instances, an object that possesses some form of energy supplies the force to do the work. In the instances described here, the objects doing the work (a student, a tractor, a pitcher, a motor/chain) possess chemical potential energy stored in food or fuel that is transformed into work. In the process of doing work, the object that is doing the work exchanges energy with the object upon which the work is done. When the work is done upon the object, that object gains energy. The energy acquired by the objects upon which work is done is known as mechanical energy.

Mechanical energy is the energy that is possessed by an object due to its motion or due to its position. Mechanical energy can be either kinetic energy (energy of motion) or potential energy (stored energy of position). Objects have mechanical energy if they are in motion and/or if they are at some position relative to a zero potential energy position (for example, a brick held at a vertical position above the ground or zero height position). A moving car possesses mechanical energy due to its motion (kinetic energy). A moving baseball possesses mechanical energy due to both its high speed (kinetic energy) and its vertical position above the ground (gravitationalpotential energy). A World Civilization book at rest on the top shelf of a locker possesses mechanical energy due to its vertical position above the ground (gravitational potential energy). A barbell lifted high above a weightlifter's head possesses mechanical energy due to its vertical position above the ground (gravitational potential energy). A drawn bow possesses mechanical energy due to its stretched position (elasticpotential energy).

The Total Mechanical Energy

As already mentioned, the mechanical energy of an object can be the result of its motion (i.e., kinetic energy) and/or the result of its stored energy of position (i.e., potential energy). The total amount of mechanical energy is merely the sum of the potential energy and the kinetic energy. This sum is simply referred to as the total mechanical energy (abbreviated TME). TME = PE + KE As discussed earlier, there are two forms of potential energy discussed in our course - gravitational potential energy and elastic potential energy. Given this fact, the above equation can be rewritten: TME = PEgrav + PEspring + KE

Power The quantity work has to do with a force causing a displacement. Work has nothing to do with the amount of time that this force acts to cause the displacement. Sometimes, the work is done very quickly and other times the work is done rather slowly. For example, a rock climber takes an abnormally long time to elevate her body up a few meters along the side of a cliff. On the other hand, a trail hiker (who selects the easier path up the mountain) might elevate her body a few meters in a short amount of time. The two people might do the same amount of work, yet the hiker does the work in considerably less time than the rock climber. The quantity that has to do with the rate at which a certain amount of work is done is known as the power. The hiker has a greater power rating than the rock climber. Power is the rate at which work is done. It is the work/time ratio. Mathematically, it is computed using the following equation.

The standard metric unit of power is the Watt. As is implied by the equation for power, a unit of power is equivalent to a unit of work divided by a unit of time. Thus, a Watt is equivalent to a Joule/second. For historical reasons, the horsepower is occasionally used to describe the power delivered by a machine. One horsepower is equivalent to approximately 750 Watts.

Most machines are designed and built to do work on objects. All machines are typically described by a power rating. The power rating indicates the rate at which that machine can do work upon other objects. Thus, the power of a machine is the work/time ratio for that particular machine. A car engine is an example of a machine that is given a power rating. The power rating relates to how rapidly the car can accelerate the car. Suppose that a 40-horsepower engine could accelerate the car from 0 mi/hr to 60 mi/hr in 16 seconds. If this were the case, then a car with four times the horsepower could do the same amount of work in one-fourth the time. That is, a 160-horsepower engine could accelerate the same car from 0 mi/hr to 60 mi/hr in 4 seconds. The point is that for the same amount of work, power and time are inversely proportional. The power equation suggests that a more powerful engine can do the same amount of work in less time.

A person is also a machine that has a power rating. Some people are more power-full than others. That is, some people are capable of doing the same amount of work in less time or more work in the same amount of time. A common physics lab involves quickly climbing a flight of stairs and using mass, height and time information to determine a student's personal power. Despite the diagonal motion along the staircase, it is often assumed that the horizontal motion is constant and all the force from the steps is used to elevate the student upward at a constant speed. Thus, the weight of the student is equal to the force that does the work on the student and the height of the staircase is the upward displacement.

PhysicsPhysics 11thSome sites for learning Physics:Physics Support: http://www.physicsclassroom.com/

Interesting Projects: http://www.sciencebuddies.org/science-fair-projects/recommender_interest_area.php?ia=Phys&dl=9

IV Quarter Exam Contents:Temperature and Thermal Equilibrium

Defining Temperature

Measuring Temperature

Temperature Conversion

Electric Charge

Properties of Electric Charge

Transfer of Electric Charge

Electric Current and Resistance

Electric Power

ContentsMonth:Magnetic Phenomena

Magnets and Magnetic Fields

Magnetism from Electricity

Magnetic Forces

Types of MagnetsThe object which contains magnetic field is called

a magnet. It has ability to attract iron, nickel etc. it was first discovered by Greek when iron is attracted by lodestone but now days, it was discovered as magnets. These are also can be formed by artificial methods in different size and shape according to their requirement. The commonly used magnet is bar magnet which is a rectangular bar in long shape which attracts ferrous objects.Similarly, compass needle is also magnet which can freely move in a horizontal direction on a pivot with one end is in North direction and the other end is in the South direction. The surrounding space of magnet feels the magnetic field. It is due to the

magnetic forcearound the magnet. So, when we place a bar magnet in a field then it will be in magnetic forces which is continuing even after removing it from magnetic field. Themagnetic fieldis produced by the flow of current. There are different types of magnets which are based on their magnetic properties and their efficiency to show magnetism.Permanent MagnetThese are types of magnets which retain certain degree of magnetism once they are magnetized. That is why they are called as permanent magnets. Some of the natural magnets which are the examples of Permanent magnets include lodestone or magnetite. A magnet always have two poles i.e. north and south poles. Suppose that we have a magnet whose length is 100 cm and it has two poles. Now if we divide this 100 cm magnet into two halves of 50 cm each, we will get two magnets each having north and south poles. If we divide this 100 cm magnet into four halves then we will get four magnets each having two poles. Therefore,a magnet with single pole never exists in nature.

field appears

to come from

a giant bar

magnet, but

with its south pole located

up near the

Earth's north pole

(near Canada).

The magnetic field lines come out of the Earth

near Antarctica and enter near Canada.

Electric Phenomena

Electric Circuits Current and Resistance

Electric Power

## Electric Field and the Movement of Charge

Perhaps one of the most useful yet taken-for-granted accomplishments of the recent centuries is the development of electric circuits. The flow of charge through wires allows us to cook our food, light our homes, air-condition our work and living space, entertain us with movies and music and even allows us to drive to work or school safely. In this unit of The Physics Classroom, we will explore the reasons for why charge flows through wires of electric circuits and the variables that affect the rate at which it flows. The means by which moving charge delivers electrical energy to appliances in order to operate them will be discussed in detail.

One of the fundamental principles that must be understood in order to grasp electric circuits pertains to the concept of how an electric field can influence charge within a circuit as it moves from one location to another. The concept of electric fieldwas first introduced in the unit on Static Electricity. In that unit, electric force was described as a non-contact force. A charged balloon can have an attractive effect upon an oppositely charged balloon even when they are not in contact. The electric force acts over the distance separating the two objects. Electric force is an action-at-a-distance force.

Action-at-a-distance forces are sometimes referred to as field forces. The concept of a

field forceis utilized by scientists to explain this rather unusual force phenomenon that occurs in the absence of physical contact. The space surrounding a charged object is affected by the presence of the charge; an electric field is established in that space. A charged object creates an electric field - an alteration of the space or field in the region that surrounds it. Other charges in that field would feel the unusual alteration of the space. Whether a charged object enters that space or not, the electric field exists. Space is altered by the presence of a charged object; other objects in that space experience the strange and mysterious qualities of the space. As another charged object enters the space and movesdeeper and deeperinto the field, the effect of the field becomes more and more noticeable.Electric field is a vector quantity whose direction is defined as the direction that a positive test charge would be pushed when placed in the field. Thus, the electric field direction about a positive source charge is always directed away from the positive source. And the electric field direction about a negative source charge is always directed toward the negative source.

Measuring Temperature

Heat and Energy

Thermal Conduction

Heat and Work

Temperature and Thermometers

What is Temperature?

Despite our built-in feel for temperature, it remains one of those concepts in science that is difficult to define. It seems that a tutorial page exploring the topic of temperature and thermometers should begin with a simple definition of temperature. But it is at this point that I'm stumped. So I turn to that familiar resource, Dictionary.com ... where I find definitions that vary from the simple-yet-not-too-enlightening to the too-complex-to-be-enlightening. At the risk of doing a belly flop in the pool of enlightenment, I will list some of those definitions here:

For certain, we are comfortable with the first two definitions - the degree or measure of how hot or cold and object is. But our understanding of temperature is not furthered by such definitions. The third and the fourth definitions that reference the kinetic energy of particles and the ability of a substance to transfer heat are scientifically accurate. However, these definitions are far too sophisticated to serve as good starting points for a discussion of temperature. So we will resign to a definition similar to the fifth one that is listed - temperature can be defined as the reading on a thermometer. Admittedly, this definition lacks the power that is needed for eliciting the much-desired Aha! Now I Understand! moment. Nonetheless it serves as a great starting point for this lesson and heat and temperature.Temperatureis what the thermometer reads. Whatever it is that temperature is a measure of, it is reflected by the reading on a thermometer.

How a Thermometer Works

Today, there are a variety of types of thermometers. The type that most of us are familiar with from science class is the type that consists of a liquid encased in a narrow glass column. Older thermometers of this type used liquid mercury. In response to our understanding of the health concerns associated with mercury exposure, these types of thermometers usually use some type of liquid alcohol. These liquid thermometers are based on the principal of thermal expansion. When a substance gets hotter, it expands to a greater volume. Nearly all substances exhibit this behavior of thermal expansion. It is the basis of the design and operation of thermometers.

As the temperature of the liquid in a thermometer increases, its volume increases. The liquid is enclosed in a tall, narrow glass (or plastic) column with a constant cross-sectional area. The increase in volume is thus due to a change in height of the liquid within the column. The increase in volume, and thus in the height of the liquid column, is proportional to the increase in temperature. Suppose that a 10-degree increase in temperature causes a 1-cm increase in the column's height. Then a 20-degree increase in temperature will cause a 2-cm increase in the column's height. And a 30-degree increase in temperature will cause s 3-cm increase in the column's height. The relationship between the temperature and the column's height is linear over the small temperature range for which the thermometer is used. This linear relationship makes the calibration of a thermometer a relatively easy task.

The calibration of any measuring tool involves the placement of divisions or marks upon the tool to measure a quantity accurately in comparison to known standards. Any measuring tool - even a meter stick - must be calibrated. The tool needs divisions or markings; for instance, a meter stick typically has markings every 1-cm apart or every 1-mm apart. These markings must be accurately placed and the accuracy of their placement can only be judged when comparing it to another object known to have an accurate length. A thermometer is calibrated by using two objects of known temperatures. The typical process involves using the freezing point and the boiling point of water. Water is known to freeze at 0°C and to boil at 100°C at an atmospheric pressure of 1 atm. By placing a thermometer in mixture of ice water and allowing the thermometer liquid to reach a stable height, the 0-degree mark can be placed upon the thermometer. Similarly, by placing the thermometer in boiling water (at 1 atm of pressure) and allowing the liquid level to reach a stable height, the 100-degree mark can be placed upon the thermometer. With these two markings placed upon the thermometer, 100 equally spaced divisions can be placed between them to represent the 1-degree marks. Since there is a linear relationship between the temperature and the height of the liquid, the divisions between 0 degree and 100 degree can be equally spaced. With a calibrated thermometer, accurate measurements can be made of the temperature of any object within the temperature range for which it has been calibrated.

Temperature Scales

The thermometer calibration process described above results in what is known as a centigrade thermometer. A centigrade thermometer has 100 divisions or intervals between the normal freezing point and the normal boiling point of water. Today, the centigrade scale is known as the Celsius scale, named after the Swedish astronomer Anders Celsius who is credited with its development. The Celsius scale is the most widely accepted temperature scale used throughout the world. It is the standard unit of temperature measurement in nearly all countries, the most notable exception being the United States. Using this scale, a temperature of 28 degrees Celsius is abbreviated as 28°C.

Traditionally slow to adopt the metric system and other accepted units of measurements, the United States more commonly uses the Fahrenheit temperature scale. A thermometer can be calibrated using the Fahrenheit scale in a similar manner as was described above. The difference is that the normal freezing point of water is designated as 32 degrees and the normal boiling point of water is designated as 212 degrees in the Fahrenheit scale. As such, there are 180 divisions or intervals between these two temperatures when using the Fahrenheit scale. The Fahrenheit scale is named in honor of German physicist Daniel Fahrenheit. A temperature of 76 degree Fahrenheit is abbreviated as 76°F. In most countries throughout the world, the Fahrenheit scale has been replaced by the use of the Celsius scale.

Temperatures expressed by the Fahrenheit scale can be converted to the Celsius scale equivalent using the equation below:

°C = (°F - 32°)/1.8

Similarly, temperatures expressed by the Celsius scale can be converted to the Fahrenheit scale equivalent using the equation below:

°F= 1.8•°C + 32°

The Kelvin Temperature Scale

While the Celsius and Fahrenheit scales are the most widely used temperature scales, there are several other scales that have been used throughout history. For example, there is the Rankine scale, the Newton scale and the Romer scale, all of which are rarely used. Finally, there is the Kelvin temperature scale, which is the standard metric system of temperature measurement and perhaps the most widely used temperature scale used among scientists. The Kelvin temperature scale is similar to the Celsius temperature scale in the sense that there are 100 equal degree increments between the normal freezing point and the normal boiling point of water. However, the zero-degree mark on the Kelvin temperature scale is 273.15 units cooler than it is on the Celsius scale. So a temperature of 0 Kelvin is equivalent to a temperature of -273.15 °C. Observe that the degree symbol is not used with this system. So a temperature of 300 units above 0 Kelvin is referred to as 300 Kelvin and not 300 degree Kelvin; such a temperature is abbreviated as 300 K. Conversions between Celsius temperatures and Kelvin temperatures (and vice versa) can be performed using one of the two equations below.

°C = K - 273.15°

K = °C + 273.15

The zero point on the Kelvin scale is known as absolute zero. It is the lowest temperature that can be achieved. The concept of an absolute temperature minimum was promoted by Scottish physicist William Thomson (a.k.a. Lord Kelvin) in 1848. Thomson theorized based on thermodynamic principles that the lowest temperature which could be achieved was -273°C. Prior to Thomson, experimentalists such as Robert Boyle (late 17th century) were well aware of the observation that the volume (and even the pressure) of a sample of gas was dependent upon its temperature. Measurements of the variations of pressure and volume with changes in the temperature could be made and plotted. Plots of volume vs. temperature (at constant pressure) and pressure vs. temperature (at constant volume) reflected the same conclusion - the volume and the pressure of a gas reduces to zero at a temperature of -273°C. Since these are the lowest values of volume and pressure that are possible, it is reasonable to conclude that -273°C was the lowest temperature that was possible.

Thomson referred to this minimum lowest temperature as absolute zero and argued that a temperature scale be adopted that had absolute zero as the lowest value on the scale. Today, that temperature scale bears his name. Scientists and engineers have been able to cool matter down to temperatures close to -273.15°C, but never below it. In the process of cooling matter to temperatures close to absolute zero, a variety of unusual properties have been observed. These properties include superconductivity, superfluidity and a state of matter known as a Bose-Einstein condensate.

Thermometers as Speedometers

Temperature is defined as the reading on a thermometer. The process of calibrating a thermometer was explained and the variety of commonly used temperature scales were described. Finally, the concept of an absolute lowest temperature was discussed. But in the end, the fundamental definition of temperature was not given. Temperature was only defined in practical terms - the reading on a thermometer. Now we have to answer the more fundamental question: what is the reading on a thermometer the reflection of? What is temperature a measure of?

It is at this point that we can use a more sophisticated definition of temperature. Temperature is a measure of the average kinetic energy of the particles within a sample of matter. Kinetic energy was defined as the energy of motion. An object ... or a particle ... that is moving has kinetic energy. There are three common forms of kinetic energy - vibrational kinetic energy, rotational kinetic energy and translational kinetic energy. Up to this point of the tutorial, we have associated kinetic energy with the movement of an object (or particle) from one location to another. This is referred to as translational kinetic energy. A ball moving through space has translational kinetic energy. An object can also have vibrational kinetic energy; this is the energy of motion possessed by an object that is oscillating or vibrating about a fixed position. A mass attached to a spring has vibrational kinetic energy. Such a mass is not permanently displaced from its position like a ball moving through space. Finally an object can have rotational kinetic energy; this is the energy associated with an object that is rotating about an imaginary axis of rotation. A spinning top isn't moving through space and isn't vibrating about a fixed position, but there is still kinetic energy associated with its motion about an axis of rotation. This form of kinetic energy is called rotational kinetic energy.

A sample of matter consists of particles that can be vibrating, rotating and moving through the space of its container. So at the particle level, a sample of matter possesses kinetic energy. A warm cup of water on a countertop may appear to be as still as can be; yet it still has kinetic energy. At the particle level, there are atoms and molecules that are vibrating, rotating and moving through the space of its container. Stick a thermometer in the cup of water and you will see the evidence that the water possesses kinetic energy. The water's temperature, as reflected by the thermometer's reading, is a measure of the average amount of kinetic energy possessed by the water molecules.

When the temperature of an object increases, the particles that compose the object begin to move faster. They either vibrate more rapidly, rotate with greater frequency or move through space with a greater speed. Increasing the temperature causes an increase in the particle speed. So as a sample of water in a pot is heated, its molecules begin to move with greater speed and this greater speed is reflected by a higher thermometer reading. Similarly, if a sample of water is placed in the freezer, its molecules begin to move slower (with a lower speed) and this is reflected by a lower thermometer reading. It is in this sense that a thermometer can be thought of as a speedometer.

What is Heat?

Consider a very hot mug of coffee on the countertop of your kitchen. For discussion purposes, we will say that the cup of coffee has a temperature of 80°C and that the surroundings (countertop, air in the kitchen, etc.) has a temperature of 26°C. What do you suppose will happen in this situation? I suspect that you know that the cup of coffee will gradually cool down over time. At 80°C, you wouldn't dare drink the coffee. Even the coffee mug will likely be too hot to touch. But over time, both the coffee mug and the coffee will cool down. Soon it will be at a drinkable temperature. And if you resist the temptation to drink the coffee, it will eventually reach room temperature. The coffee cools from 80°C to about 26°C. So what is happening over the course of time to cause the coffee to cool down? The answer to this question can be both macroscopic and particulate in nature.

On the macroscopic level, we would say that the coffee and the mug are transferring heat to the surroundings. This transfer of heat occurs from the hot coffee and hot mug to the surrounding air. The fact that the coffee lowers its temperature is a sign that the average kinetic energy of its particles is decreasing. The coffee is losing energy. The mug is also lowering its temperature; the average kinetic energy of its particles is also decreasing. The mug is also losing energy. The energy that is lost by the coffee and the mug is being transferred to the colder surroundings. We refer to this transfer of energy from the coffee and the mug to the surrounding air and countertop as heat. In this sense, heat is simply the transfer of energy from a hot object to a colder object.

Now let's consider a different scenario - that of a cold can of pop placed on the same kitchen counter. For discussion purposes, we will say that the pop and the can which contains it has a temperature of 5°C and that the surroundings (countertop, air in the kitchen, etc.) has a temperature of 26°C. What will happen to the cold can of pop over the course of time? Once more, I suspect that you know the answer. The cold pop and the container will both warm up to room temperature. But what is happening to cause these colder-than-room-temperature objects to increase their temperature? Is the cold escaping from the pop and its container? No! There is no such thing as the cold escaping orleaking. Rather, our explanation is very similar to the explanation used to explain why the coffee cools down. There is a heat transfer.

Over time, the pop and the container increase their temperature. The temperature rises from 5°C to nearly 26°C. This increase in temperature is a sign that the average kinetic energy of the particles within the pop and the container is increasing. In order for the particles within the pop and the container to increase their kinetic energy, they must be gaining energy from somewhere. But from where? Energy is being transferred from the surroundings (countertop, air in the kitchen, etc.) in the form of heat. Just as in the case of the cooling coffee mug, energy is being transferred from the higher temperature objects to the lower temperature object. Once more, this is known as heat - the transfer of energy from the higher temperature object to a lower temperature object.

Both of these scenarios could be summarized by two simple statements. An object decreases its temperature by releasing energy in the form of heat to its surroundings. And an object increases its temperature by gaining energy in the form of heat from its surroundings. Both the warming upand the cooling down of objects works in the same way - by heat transfer from the higher temperature object to the lower temperature object. So now we can meaningfully re-state the definition of temperature. Temperature is a measure of the ability of a substance, or more generally of any physical system, to transfer heat energy to another physical system. The higher the temperature of an object is, the greater the tendency of that object to transfer heat. The lower the temperature of an object is, the greater the tendency of that object to be on the receiving end of the heat transfer.

But perhaps you have been asking: what happens to the temperature of surroundings? Do the countertop and the air in the kitchen increase their temperature when the mug and the coffee cool down? And do the countertop and the air in the kitchen decrease its temperature when the can and its pop warm up? The answer is a resounding Yes! The proof? Just touch the countertop - it should feel cooler or warmer than before the coffee mug or pop can were placed on the countertop. But what about the air in the kitchen? Now that's a little more difficult to present a convincing proof of. The fact that the volume of air in the room is so large and that the energy quickly diffuses away from the surface of the mug and of the means that the temperature change of the air in the kitchen will be abnormally small. In fact, it will be negligibly small. There would have to be a lot more heat transfer before there is a noticeable temperature change.

Thermal Equilibrium

In the discussion of the cooling of the coffee mug, the countertop and the air in the kitchen were referred to as thesurroundings. It is common in physics discussions of this type to use a mental framework of a system and thesurroundings. The coffee mug (and the coffee) would be regarded as the system and everything else in the universe would be regarded as the surroundings. To keep it simple, we often narrow the scope of the surroundings from the rest of the universe to simply those objects that are immediately surrounding the system. This approach of analyzing a situation in terms of system and surroundings is so useful that we will adopt the approach for the rest of this chapter and the next.

Now let's imagine a third situation. Suppose that a small metal cup of hot water is placed inside of a larger Styrofoam cup of cold water. Let's suppose that the temperature of the hot water is initially 70°C and that the temperature of the cold water in the outer cup is initially 5°C. And let's suppose that both cups are equipped with thermometers (or temperature probes) that measure the temperature of the water in each cup over the course of time. What do you suppose will happen? Before you read on, think about the question and commit to some form of answer. When the cold water is done warming and the hot water is done cooling, will their temperatures be the same or different? Will the cold water warm up to a lower temperature than the temperature that the hot water cools down to? Or as the warming and cooling occurs, will their temperatures cross each other?

As you can see from the graph, the hot water cooled down to approximately 30°C and the cold water warmed up to approximately the same temperature. Heat is transferred from the high temperature object (inner can of hot water) to the low temperature object (outer can of cold water). If we designate the inner cup of hot water as the system, then we can say that there is a flow of water from the system to the surroundings. As long as there is a temperature difference between the system and the surroundings, there is a heat flow between them. The heat flow is more rapid at first as depicted by the steeper slopes of the lines. Over time, the temperature difference between system and surroundings decreases and the rate of heat transfer decreases. This is denoted by the gentler slope of the two lines. (Detailed information about rates of heat transfer will be discussed later in this lesson.) Eventually, the system and the surroundings reach the same temperature and the heat transfer ceases. It is at this point, that the two objects are said to have reached thermal equilibrium.

Heat Changes the Temperature of Objects

What does heat do? First, it changes the temperature of an object. If heat is transferred from an object to the surroundings, then the object can cool down and the surroundings can warm up. When heat is transferred to an object by its surroundings, then the object can warm up and the surroundings can cool down. Heat, once absorbed as energy, contributes to the overall internal energy of the object. One form of this internal energy is kinetic energy; the particles begin to move faster, resulting in a greater kinetic energy. This more vigorous motion of particles is reflected by a temperature increase. The reverse logic applies as well. Energy, once released as heat, results in a decrease in the overall internal energy of the object. Since kinetic energy is one of the forms of internal energy, the release of heat from an object causes a decrease in the average kinetic energy of its particles. This means that the particles move more sluggishly and the temperature of the object decreases. The release or absorption of energy in the form heat by an object is often associated with a temperature change of that object. This was the focus of the Thermometers as Speedometers. What can be said of the object can also be said of the surroundings. The release or absorption of energy in the form heat by the surroundings is often associated with a temperature change of the surroundings. We often find that the transfer of heat causes a temperature change in both system and surroundings. One warms up and the other cools down.

Heat Changes the State of Matter

But does the absorption or release of energy in the form of heat always cause a temperature change? Surprisingly, the answer is no. To illustrate why, consider the following situation, which is often demonstrated or even experimented with in a thermal physics unit in school. Para-dichlorobenzene, the main ingredient in many forms of mothballs, has a melting point of about 54 °C. Suppose that a sample of the chemical is collected in a test tube and heated to about 80°C. The para-dichlorobenzene will be in the liquid state (though much of it will have sublimed and be filling the room with a most noticeable aroma). Now suppose that a thermometer is inserted in the test tube and that the test tube is placed in a beaker of room temperature water. Temperature-time data can be collected every 10 seconds. Quite expectedly, one notices that the temperature of the para-dichlorobenzene gradually decreases. As heat is transferred from the high temperature test tube to the low temperature water, the temperature of the liquid para-dichlorbenzene decreases. But then quite unexpectedly, one would notice that this steady decrease in temperature ceases at about 54°C. Once the temperature of liquid para-dichlorbenzene decreases to 54°C, the thermometer level suddenly stands still. Based on the thermometer reading, you might think that no heat was being transferred. But a look in the test tube reveals dramatic change taking place. The liquid para-dichlorbenzene is crystallizing to form solid para-dichlorbenzene. Once the last trace of liquid para-dichlorbenzene vanishes (and it is in all solid form), the temperature begins to decrease again from 54°C to the temperature of the water. How can these observations help us to understand the question of what does heat do?

First, the decrease in temperature from 80°C to 54°C is easy to explain. Heat is transferred between two adjacent objects that are at different temperatures. The test tube and the para-dichlorbenzene are at a higher temperature than the surrounding water of the beaker. Heat will flow from the test tube of para-dichlorbenzene to the water, causing the para-dichlorbenzene to cool down and the water to warm up. And the decrease in temperature from 54°C to the temperature of the water in the beaker is also easily explainable. Two adjacent objects of different temperatures will transfer heat between them until thermal equilibrium is reached. The difficult explanation involves explaining what happens at 54°C. Why does the temperature no longer decrease when the liquid para-dichlorbenzene begins to crystallize? Is there still a transfer of heat between the test tube of para-dichlorbenzene and the beaker of water even when the temperature isn't changing?

The answer to the question Is heat being transferred? is a resounding yes! After all, the principle is that heat is always transferred between two adjacent objects that are at different temperatures. A thermometer placed in the water reveals that the water is still warming up even though there is no temperature change in the para-dichlorbenzene. So heat is definitely being transferred from the para-dichlorbenzene to the water. But why does the temperature of the para-dichlorbenzene remain constant during this crystallization period? Before the para-dichlorbenzene can continue to lower its temperature, it must first transition from the liquid state to the solid state. The crystallization of para-dichlorbenzene occurs at 54°C - the freezing point of the substance. At this temperature, the energy that is lost by the para-dichlorbenzene is associated with a change in the other form of internal energy - potential energy. A substance not only possesses kinetic energy due to the motion of its particles, it also possesses potential energy due to the intermolecular attractions between particles. As the para-dichlorbenzene crystallizes at 54°C, the energy being lost is reflected by decreases in the potential energy of the para-dichlorbenzene as it changes state. Once all the para-dichlorbenzene has changed to the solid state, the loss of energy is once more reflected by a decrease in the kinetic energy of the substance; its temperature decreases.

Hooke’s Law

The Simple Pendulum

Amplitude, Period, Frequency

Wave Types and Motion

Period, Frequency and Wave Speed

Vibrational Motion

Things wiggle. They do the back and forth. They vibrate; they shake; they oscillate. These phrases describe the motion of a variety of objects. They even describe the motion of matter at the atomic level. Even atoms wiggle - they do the back and forth. Wiggles, vibrations, and oscillations are an inseparable part of nature. In this chapter of The Physics Classroom Tutorial, we will make an effort to understand vibrational motion and its relationship to waves. An understanding of vibrations and waves is essential to understanding our physical world. Much of what we see and hear is only possible because of vibrations and waves. We see the world around us because of light waves. And we hear the world around us because of sound waves. If we can understand waves, then we will be able to understand the world of sight and sound.

To begin our ponderings of vibrations and waves, consider one of those crazy bobblehead dolls that you've likely seen at baseball stadiums or novelty shops. A bobblehead doll consists of an oversized replica of a person's head attached by a spring to a body and a stand. A light tap to the oversized head causes it to bobble. The head wiggles; it vibrates; it oscillates. When pushed or somehow disturbed, the head does the back and forth. The back and forth doesn't happen forever. Over time, the vibrations tend to die off and the bobblehead stops bobbing and finally assumes its usual resting position.

The bobblehead doll is a good illustration of many of the principles of vibrational motion. Think about how you would describe the back and forth motion of the oversized head of a bobblehead doll. What words would you use to describe such a motion? How does the motion of the bobblehead change over time? How does the motion of one bobblehead differ from the motion of another bobblehead? What quantities could you measure to describe the motion and so distinguish one motion from another motion? How would you explain the cause of such a motion? Why does the back and forth motion of the bobblehead finally stop? These are all questions worth pondering and answering if we are to understand vibrational motion.

What Causes Objects to Vibrate?

Like any object that undergoes vibrational motion, the bobblehead has a resting position. The resting position is the position assumed by the bobblehead when it is not vibrating. The resting position is sometimes referred to as theequilibrium position. When an object is positioned at its equilibrium position, it is in a state of equilibrium. An object which is in a state of equilibrium is experiencing a balance of forces. All the individual forces - gravity, spring, etc. - are balanced or add up to an overall net force of 0 Newtons. When a bobblehead is at the equilibrium position, the forces on the bobblehead are balanced. The bobblehead will remain in this position until somehow disturbed from its equilibrium.

If a force is applied to the bobblehead, the equilibrium will be disturbed and the bobblehead will begin vibrating. We could use the phrase forced vibration to describe the force which sets the otherwise resting bobblehead into motion. In this case, the force is a short-lived, momentary force that begins the motion. The bobblehead does its back and forth, repeating the motion over and over. Each repetition of its back and forth motion is a little less vigorous than its previous repetition. If the head sways 3 cm to the right of its equilibrium position during the first repetition, it may only sway 2.5 cm to the right of its equilibrium position during the second repetition. And it may only sway 2.0 cm to the right of its equilibrium position during the third repetition. And so on. The extent of its displacement from the equilibrium position becomes less and less over time. Because the forced vibration that initiated the motion is a single instance of a short-lived, momentary force, the vibrations ultimately cease. The bobblehead is said to experiencedamping. Damping is the tendency of a vibrating object to lose or to dissipate its energy over time. The mechanical energy of the bobbing head is lost to other objects. Without a sustained forced vibration, the back and forth motion of the bobblehead eventually ceases as energy is dissipated to other objects. A sustained input of energy would be required to keep the back and forth motion going. After all, if the vibrating object naturally loses energy, then it must continuously be put back into the system through a forced vibration in order to sustain the vibration.

The Restoring Force

A vibrating bobblehead often does the back and forth a number of times. The vibrations repeat themselves over and over. As such, the bobblehead will move back to (and past) the equilibrium position every time it returns from its maximum displacement to the right or the left (or above or below). This begs a question - and perhaps one that you have been thinking of yourself as you've pondered the topic of vibration. If the forces acting upon the bobblehead are balanced when at the equilibrium position, then why does the bobblehead sway past this position? Why doesn't the bobblehead stop the first time it returns to the equilibrium position? The answer to this question can be found in Newton's first law of motion. Like any moving object, the motion of a vibrating object can be understood in light of Newton's laws. According to Newton's law of inertia, an object which is moving will continue its motion if the forces are balanced. Put another way, forces, when balanced, do not stop moving objects. So every instant in time that the bobblehead is at the equilibrium position, the momentary balance of forces will not stop the motion. The bobblehead keeps moving. It moves past the equilibrium position towards the opposite side of its swing. As the bobblehead is displaced past its equilibrium position, then a force capable of slowing it down and stopping it exists. This force that slows the bobblehead down as it moves away from its equilibrium position is known as a restoring force. The restoring force acts upon the vibrating object to move it back to its original equilibrium position.

Vibrational motion is often contrasted with translational motion. In translational motion, an object is permanently displaced. The initial force that is imparted to the object displaces it from its resting position and sets it into motion. Yet because there is no restoring force, the object continues the motion in its original direction. When an object vibrates, it doesn't move permanently out of position. The restoring force acts to slow it down, change its direction and force it back to its original equilibrium position. An object in translational motion is permanently displaced from its original position. But an object in vibrational motion wiggles about a fixed position - its original equilibrium position. Because of the restoring force, vibrating objects do the back and forth.

Other Vibrating Systems

As you know, bobblehead dolls are not the only objects that vibrate. It might be safe to say that all objects in one way or another can be forced to vibrate to some extent. The vibrations might not be large enough to be visible. Or the amount of damping might be so strong that the object scarcely completes a full cycle of vibration. But as long as a force persists to restore the object to its original position, a displacement from its resting position will result in a vibration. Even a large massive skyscraper is known to vibrate as winds push upon its structure. While held fixed in place at its foundation (we hope), the winds force the length of the structure out of position and the skyscraper is forced into vibration.

A pendulum is a classic example of an object that is considered to vibrate. A simple pendulum consists of a relatively massive object hung by a string from a fixed support. It typically hangs vertically in its equilibrium position. When the mass is displaced from equilibrium, it begins its back and forth vibration about its fixed equilibrium position. The motion is regular and repeating. In the next part of this lesson, we will describe such a regular and repeating motion as a periodic motion. Because of the regular nature of a pendulum's motion, many clocks, such as grandfather clocks, use a pendulum as part of its timing mechanism.

An inverted pendulum is another classic example of an object that undergoes vibrational motion. An inverted pendulum is simply a pendulum which has its fixed end located below the vibrating mass. An inverted pendulum can be made by attaching a mass (such as a tennis ball) to the top end of a dowel rod and then securing the bottom end of the dowel rod to a horizontal support. This is shown in the diagram below. A gentle force exerted upon the tennis ball will cause it to vibrate about a fixed, equilibrium position. The vibrating skyscraper can be thought of as a type of inverted pendulum. Tall trees are often displaced from their usual vertical orientation by strong winds. As the winds cease, the trees will vibrate back and forth about their fixed positions. Such trees can be thought of as acting as inverted pendula. Even the tines of a tuning fork can be considered a type of inverted pendulum

Another classic example of an object that undergoes vibrational motion is a mass on a spring. The animation at the right depicts a mass suspended from a spring. The mass hangs at a resting position. If the mass is pulled down, the spring is stretched. Once the mass is released, it begins to vibrate. It does the back and forth, vibrating about a fixed position. If the spring is rotated horizontally and the mass is placed upon a supporting surface, the same back and forth motion can be observed. Pulling the mass to the right of its resting position stretches the spring. When released, the mass is pulled back to the left, heading towards its resting position. After passing by its resting position, the spring begins to compress. The compressions of the coiled spring result in a restoring force that again pushes rightward on the leftward moving mass. The cycle continues as the mass vibrates back and forth about a fixed position. The springs inside of a bed mattress, the suspension systems of some cars, and bathroom scales all operated as a mass on a spring system.

In all the vibrating systems just mentioned, damping is clearly evident. The simple pendulum doesn't vibrate forever; its energy is gradually dissipated through air resistance and loss of energy to the support. The inverted pendulum consisting of a tennis ball mounted to the top of a dowel rod does not vibrate forever. Like the simple pendulum, the energy of the tennis ball is dissipated through air resistance and vibrations of the support. Frictional forces also cause the mass on a spring to lose its energy to the surroundings. In some instances, damping is a favored feature. Car suspension systems are intended to dissipate vibrational energy, preventing drivers and passengers from having to do the back and forth as they also do the down the road.

Hopefully a lot of our original questions have been answered. But one question that has not yet been answered is the question pertaining to quantities that can be measured. How can we quantitatively describe a vibrating object? What measurements can be made of vibrating objects that would distinguish one vibrating object from another? We will ponder this question in the next part of this lesson on vibrational motion.

Properties of Periodic Motion

A vibrating object is wiggling about a fixed position. Like the mass on a spring in the animation at the right, a vibrating object is moving over the same path over the course of time. Its motion repeats itself over and over again. If it were not for damping, the vibrations would endure forever (or at least until someone catches the mass and brings it to rest). The mass on the spring not only repeats the same motion, it does so in a regular fashion. The time it takes to complete one back and forth cycle is always the same amount of time. If it takes the mass 3.2 seconds for the mass to complete the first back and forth cycle, then it will take 3.2 seconds to complete the seventh back and forth cycle. It's like clockwork. It's so predictable that you could set your watch by it. In Physics, a motion that is regular and repeating is referred to as a periodic motion. Most objects that vibrate do so in a regular and repeated fashion; their vibrations are periodic.

One obvious characteristic of the graph has to do with its shape. Many students recognize the shape of this graph from experiences in Mathematics class. The graph has the shape of a sine wave. If y = sine(x) is plotted on a graphing calculator, a graph with this same shape would be created. The vertical axis of the above graph represents the position of the mass relative to the motion detector. A position of about 0.60 m cm above the detector represents the resting position of the mass. So the mass is vibrating back and forth about this fixed resting position over the course of time. There is something sinusoidal about the vibration of a mass on a spring. And the same can be said of a pendulum vibrating about a fixed position or of a guitar string or of the air inside of a wind instrument. The position of the mass is a function of the sine of the time.

A second obvious characteristic of the graph may be its periodic nature. The motion repeats itself in a regular fashion. Time is being plotted along the horizontal axis; so any measurement taken along this axis is a measurement of the time for something to happen. A full cycle of vibration might be thought of as the movement of the mass from its resting position (A) to its maximum height (B), back down past its resting position (C) to its minimum position (D), and then back to its resting position (E). Using measurements from along the time axis, it is possible to determine the time for one complete cycle. The mass is at position A at a time of 0.0 seconds and completes its cycle when it is at position E at a time of 2.3 seconds. It takes 2.3 seconds to complete the first full cycle of vibration. Now if the motion of this mass is periodic (i.e., regular and repeating), then it should take the same time of 2.3 seconds to complete any full cycle of vibration. The same time-axis measurements can be taken for the sixth full cycle of vibration. In the sixth full cycle, the mass moves from a resting position (U) up to V, back down past W to X and finally back up to its resting position (Y) in the time interval from 11.6 seconds to 13.9 seconds. This represents a time of 2.3 seconds to complete the sixth full cycle of vibration. The two cycle times are identical. Other cycle times are indicated in the table below. By inspection of the table, one can safely conclude that the motion of the mass on a spring is regular and repeating; it is clearly periodic. The small deviation from 2.3 s in the third cyle can be accounted for by the lack of precision in the reading of the graph.

A third obvious characteristic of the graph is that damping occurs with the mass-spring system. Some energy is being dissipated over the course of time. The extent to which the mass moves above (B, F, J, N, R and V) or below (D, H, L, P, T and X) the resting position (C, E, G, I, etc.) varies over the course of time. In the first full cycle of vibration being shown, the mass moves from its resting position (A) 0.60 m above the motion detector to a high position (B) of 0.99 m cm above the motion detector. This is a total upward displacement of 0.29 m. In the sixth full cycle of vibration that is shown, the mass moves from its resting position (U) 0.60 m above the motion detector to a high position (V) 0.94 m above the motion detector. This is a total upward displacement of 0.24 m cm. The table below summarizes displacement measurements for several other cycles displayed on the graph.

- the time for the mass to complete a cycle, and
- the maximum displacement of the mass above (or below) the resting position.

These two measurable quantities have names. We call these quantities period and amplitude.Period and Frequency

An object that is in periodic motion - such as a mass on a spring, a pendulum or a bobblehead doll - will undergo back and forth vibrations about a fixed position in a regular and repeating fashion. The fact that the periodic motion is regular and repeating means that it can be mathematically described by a quantity known as the period. The period of the object's motion is defined as the time for the object to complete one full cycle. Being a time, the period is measured in units such as seconds, milliseconds, days or even years. The standard metric unit for period is the second.

An object in periodic motion can have a long period or a short period. For instance, a pendulum bob tied to a 1-meter length string has a period of about 2.0 seconds. For comparison sake, consider the vibrations of a piano string that plays themiddle C note (the C note of the fourth octave). Its period is approximately 0.0038 seconds (3.8 milliseconds). When comparing these two vibrating objects - the 1.0-meter length pendulum and the piano string which plays the middle C note - we would describe the piano string as vibrating relatively frequently and we would describe the pendulum as vibrating relatively infrequently. Observe that the description of the two objects uses the termsfrequently and infrequently. The terms fast and slow are not used since physics types reserve the words fast and slow to refer to an object's speed. Here in this description we are referring to the frequency, not the speed. An object can be in periodic motion and have a low frequency and a high speed. As an example, consider the periodic motion of the moon in orbit about the earth. The moon moves very fast; its orbit is highly infrequent. It moves through space with a speed of about 1000 m/s - that's fast. Yet it makes a complete cycle about the earth once every 27.3 days (a period of about 2.4x105 seconds) - that's infrequent.

The concept and quantity frequency is best understood if you attach it to the everyday English meaning of the word. Frequency is a word we often use to describe how often something occurs. You might say that you frequently check your email or you frequently talk to a friend or you frequently wash your hands when working with chemicals. Used in this context, you mean that you do these activities often. To say that you frequently check your email means that you do it several times a day - you do it often. In physics, frequency is used with the same meaning - it indicates how often a repeated event occurs. High frequency events that are periodic occur often, with little time in between each occurrence - like the back and forth vibrations of the tines of a tuning fork. The vibrations are so frequent that they can't be seen with the naked eye. A 256-Hz tuning fork has tines that make 256 complete back and forth vibrations each second. At this frequency, it only takes the tines about 0.00391 seconds to complete one cycle. A 512-Hz tuning fork has an even higher frequency. Its vibrations occur more frequently; the time for a full cycle to be completed is 0.00195 seconds. In comparing these two tuning forks, it is obvious that the tuning fork with the highest frequency has the lowest period. The two quantities frequency and period are inversely related to each other. In fact, they are mathematical reciprocals of each other. The frequency is the reciprocal of the period and the period is the reciprocal of the frequency.

This reciprocal relationship is easy to understand. After all, the two quantities are conceptual reciprocals (a phrase I made up). Consider their definitions as restated below:

Amplitude of Vibration: The final measurable quantity that describes a vibrating object is the amplitude. The amplitude is defined as the maximum displacement of an object from its resting position. The resting position is that position assumed by the object when not vibrating. Once vibrating, the object oscillates about this fixed position. If the object is a mass on a spring (such as the discussion earlier on this page), then it might be displaced a maximum distance of 35 cm below the resting position and 35 cm above the resting position. In this case, the amplitude of motion is 35 cm.

Over the course of time, the amplitude of a vibrating object tends to become less and less. The amplitude of motion is a reflection of the quantity of energy possessed by the vibrating object. An object vibrating with a relatively large amplitude has a relatively large amount of energy. Over time, some of this energy is lost due to damping. As the energy is lost, the amplitude decreases. If given enough time, the amplitude decreases to 0 as the object finally stops vibrating. At this point in time, it has lost all its energy.

Fluids

Density and Buoyant Force

Archimedes Principle

Fluid Pressure

Fluids

A

fluidis a substance that cannot maintain its own shape but takes the shape of its container.Fluid laws assume idealized fluids that cannot be compressed.

Density and pressure

The density (ρ) of a substance of uniform composition is its mass per unit volume: ρ = m/ V. In the SI system, density is measured in units of kilograms per cubic meter. Imagine an upright cylindrical beaker filled with a fluid. The fluid exerts a force on the bottom of the container due to its weight. Pressure is defined as the force per unit area: P

F/ A, or in terms of magnitude, P

mg/A, wheremgis the weight of the fluid. The SI unit of pressure is N/m2, called a pascal. The pressure at the bottom of a fluid can be expressed in terms of the density (ρ) and height(h)of the fluid:or

P= ρhg. The pressure at any point in a fluid acts equally in all directions. This concept is sometimes called thebasic law of fluid pressure.Archimedes' principle

Water commonly provides partial support for any object placed in it. The upward force on an object placed in a fluid is called the buoyant force. According to Archimedes' principle, the magnitude of a buoyant force on a completely or partially submerged object always equals the weight of the fluid displaced by the object.

Archimedes' principle can be verified by a nonmathematical argument. Consider the cubic volume of water in the container of water shown in Figure 2 . This volume is in equilibrium with the forces acting on it, which are the weight and the buoyant force; therefore, the downward force of the weight ( W) must be balanced by the upward buoyant force ( B), which is provided by the rest of the water in the container.

If a solid floats partially submerged in a liquid, the volume of liquid displaced is less than the volume of the solid. Comparing the density of the solid and the liquid in which it floats leads to an interesting result. The formulas for density are Ds = ms/ Vs and Dl = ml/ Vl, where D is the density, V is the volume, m is the mass, and the subscripts s and l refer to quantities associated with the solid and the liquid respectively. Solving for the masses leads to ms = DsVs and ml = DlVl. According to Archimedes' principle, the weights of the solid and the displaced liquid are equal. Because the weights are simply mass times a constant (g), the masses must be equal also; therefore, DsVs= DlVl or Ds/Dl = VlVl. Now, V = Ah, where A is the cross-sectional area and h is the height. For a solid floating in liquid, Al=As and hl is the height of the solid that is submerged, hsub. With these substitutions, the above relationship becomes Ds/Dl = hsub/hs; therefore, the fractional part of the solid that is submerged is equal to the ratio of the density of the solid to the density of the surrounding liquid in which it floats. For example, about 90 percent of an iceberg is beneath the surface of sea water because the density of ice is about nine-tenths that of sea water.

Pascal's principle

P

ascal's principlemay be stated thus: The pressure applied at one point in an enclosed fluid under equilibrium conditions is transmitted equally to all parts of the fluid. This rule is utilized in hydraulic systems.Let the subscripts

aandbdenote the quantities at each piston. The pressures are equal; therefore,Pa=Pb. Substitute the expression for pressure in terms of force and area to obtainfa/Aa= (Fb/Ab). Substitute πr2 for the area of a circle, simplify, and solve forFb:Fb=(Fa)(rb2/ra2). Because the force exerted at pointais multiplied by the square of the ratio of the radii andrb>ra, a modest force on the small pistonacan lift a relatively larger weight on piston b.Bernoulli's equation

Imagine a fluid flowing through a section of pipe with one end having a smaller cross-sectional area than the pipe at the other end. The flow of liquids is very complex; therefore, this discussion will assume conditions of the smooth flow of an incompressible fluid through walls with no drag. The velocity of the fluid in the constricted end must be greater than the velocity at the larger end if steady flow is maintained; that is, the volume passing per time is the same at all points. Swiftly moving fluids exert less pressure than slowly moving fluids. Bernoulli's equation applies conservation of energy to formalize this observation: P + (1/2) ρ v2 + ρ gh = a constant. The equation states that the sum of the pressure (P), the kinetic energy per unit volume, and the potential energy per unit volume have the same value throughout the pipe.

Rotational motion

Torque Analysis and addition

Simple Machines

TorqueFigure 1 Definitions

= r x F = r F sin().

In other words, torque is the cross product between the distance vector (the distance from the pivot point to the point where force is applied) and the force vector, 'a' being the angle between r and F.

Using the right hand rule, we can find the direction of the torque vector. If we put our fingers in the direction of r, and curl them to the direction of F, then the thumb points in the direction of the torque vector.

Imagine pushing a door to open it. The force of your push (F) causes the door to rotate about its hinges (the pivot point, O). How hard you need to push depends on the distance you are from the hinges (r) (and several other things, but let's ignore them now). The closer you are to the hinges (i.e. the smaller r is), the harder it is to push. This is what happens when you try to push open a door on the wrong side. The torque you created on the door is smaller than it would have been had you pushed the correct side (away from its hinges).

Note that the force applied, F, and the moment arm, r, are independent of the object. Furthermore, a force applied at the pivot point will cause no torque since the moment arm would be zero (r = 0).

Let the force acting on an object be broken up into its tangential (Ftan) and radial (Frad) components (see Figure 2). (Note that the tangential component isperpendicular to the moment arm, while the radial component is parallel to the moment arm.) The radial component of the force has no contribution to the torque because it passes through the pivot point. So, it is only the tangential component of the force which affects torque (since it is perpendicular to the line between the point of action of the force and the pivot point).

Figure 2 Tangential and radial components of force F

Rotational Equilibrium is analogous to translational equilibrium, where the sum of the forces are equal to zero. In rotational equilibrium, the sum of the torques is equal to zero. In other words, there is no net torque on the object.

Note that the SI units of torque is a Newton-metre, which is also a way of expressing a Joule (the unit for energy). However, torque is not energy. So, to avoid confusion, we will use the units N.m, and not J. The distinction arises because energy is a scalar quanitity, whereas torque is a vector.

Centripetal Acceleration and Force

Gravitational Force and its applications

Speed and VelocityThe same concepts and principles used to describe and explain the motion of an object can be used to describe and explain the parabolic motion of a projectile. In this unit, we will see that these same concepts and principles can also be used to describe and explain the motion of objects that either move in circles or can be approximated to be moving in circles. Kinematic concepts and motion principles will be applied to the motion of objects in circles and then extended to analyze the motion of such objects as roller coaster cars, a football player making a

circularturn, and a planet orbiting the sun. We will see that the beauty and power of physics lies in the fact that a few simple concepts and principles can be used to explain the mechanics of the entire universe. Lesson 1 of this study will begin with the development of kinematic and dynamic ideas that can be used to describe and explain the motion of objects in circles.Uniform circular motionis the motion of an object in a circle with a constant or uniform speed.Calculation of the Average Speed

Uniform circular motion - circular motion at a constant speed - is one of many forms of circular motion. An object moving in uniform circular motion would cover the same linear distance in each second of time. When moving in a circle, an object traverses a distance around the perimeter of the circle. So if your car were to move in a circle with a constant speed of 5 m/s, then the car would travel 5 meters along the perimeter of the circle in each second of time. The distance of one complete cycle around the perimeter of a circle is known as the

circumference. With a uniform speed of 5 m/s, a car could make a complete cycle around a circle that had a circumference of 5 meters. At this uniform speed of 5 m/s, each cycle around the 5-m circumference circle would require 1 second. At 5 m/s, a circle with a circumference of 20 meters could be made in 4 seconds; and at this uniform speed, every cycle around the 20-m circumference of the circle would take the same time period of 4 seconds. This relationship between the circumference of a circle, the time to complete one cycle around the circle, and the speed of the object is merely an extension of the average speed equation. The circumference of any circle can be computed using from the radius according to the equation.Combining these two equations above will lead to a new equation relating the speed of an object moving in uniform circular motion to the radius of the circle and the time to make one cycle around the circle (

period).Rrepresents the radius of the circle andTrepresents the period. This equation, like all equations, can be used as an algebraic recipe for problem solving. It also can be used to guide our thinking about the variables in the equation relate to each other. For instance, the equation suggests that for objects moving around circles of different radius in the same period, the object traversing the circle of larger radius must be traveling with the greatest speed. In fact, the average speed and the radius of the circle are directly proportional. A twofold increase in radius corresponds to a twofold increase in speed; a threefold increase in radius corresponds to a three--fold increase in speed; and so on. To illustrate, consider a strand of four LED lights positioned at various locations along the strand. The strand is held at one end and spun rapidly in a circle. Each LED light traverses a circle of different radius. Yet since they are connected to the same wire, their period of rotation is the same. Subsequently, the LEDs that are further from the center of the circle are traveling faster in order to sweep out the circumference of the larger circle in the same amount of time. If the room lights are turned off, the LEDs created an arc that could be perceived to be longer for those LEDs that were traveling faster - the LEDs with the greatest radius. This is illustrated in the diagram at the right.The Direction of the Velocity Vector

Speed is a scalar quantity and velocity is a vector quantity. Velocity, being a vector, has both a magnitude and a direction. The magnitude of the velocity vector is the instantaneous speed of the object. The direction of the velocity vector is directed in the same direction that the object moves. Since an object is moving in a circle, its direction is continuously changing. At one moment, the object is moving northward such that the velocity vector is directed northward. One quarter of a cycle later, the object would be moving eastward such that the velocity vector is directed eastward. As the object

roundsthe circle, the direction of the velocity vector is different than it was the instant before. So while the magnitude of the velocity vector may be constant, the direction of the velocity vector is changing. The best word that can be used to describe the direction of the velocity vector is the wordtangential. The direction of the velocity vector at any instant is in the direction of a tangent line drawn to the circle at the object's location. (A tangent line is a line that touches a circle at one point but does not intersect it.) The diagram at the right shows the direction of the velocity vector at four different points for an object moving in a clockwise direction around a circle. While the actual direction of the object (and thus, of the velocity vector) is changing, its direction is always tangent to the circle.To summarize, an object moving in uniform circular motion is moving around the perimeter of the circle with a constant speed. While the speed of the object is constant, its velocity is changing. Velocity, being a vector, has a constant magnitude but a changing direction. The direction is always directed tangent to the circle and as the object turns the circle, the tangent line is always pointing in a new direction.

AccelerationAn object moving in uniform circular motion is moving in a circle with a uniform or constant speed. The velocity vector is constant in magnitude but changing in direction. Because the speed is constant for such a motion, many students have the misconception that there is no acceleration. "After all," they might say, "if I were driving a car in a circle at a constant speed of 20 mi/hr, then the speed is neither decreasing nor increasing; therefore there must not be an acceleration." At the center of this common student misconception is the wrong belief that acceleration has to do with speed and not with velocity. But the fact is that an accelerating object is an object that is changing its velocity. And since velocity is a vector that has both magnitude and direction, a change in either the magnitude or the direction constitutes a change in the velocity. For this reason, it can be safely concluded that an object moving in a circle at constant speed is indeed accelerating. It is accelerating because the direction of the velocity vector is changing.

To understand this at a deeper level, we will have to combine the definition of acceleration with a review of some basic vector principles. Acceleration as a quantity was defined as the rate at which the velocity of an object changes. As such, it is calculated using the following equation:

virepresents the initial velocity andvfrepresents the final velocity after some time oft. The numerator of the equation is found by subtracting one vector (vi) from a second vector (vf). But the addition and subtraction of vectors from each other is done in a manner much different than the addition and subtraction of scalar quantities. Consider the case of an object moving in a circle about point C as shown in the diagram below. In a time of t seconds, the object has moved from point A to point B. In this time, the velocity has changed fromvitovf. The process of subtractingvifromvfis shown in the vector diagram; this process yields the change in velocity.The Centripetal Force RequirementAn object moving in a circle is experiencing an acceleration. Even if moving around the perimeter of the circle with a constant speed, there is still

a change in velocity and subsequently an acceleration. This acceleration is directed towards the center of the circle. And in accord with Newton's second law of motion , an object which experiences an acceleration must also be experiencing a net force. The direction of the net force is in the same direction as the acceleration. So for an object moving in a circle, there must be an inward force acting upon it in order to cause its inward acceleration. This is sometimes referred to as the

centripetal force requirement. The wordcentripetal(not to be confused with the F-wordcentrifugal) means center seeking. For object's moving in circular motion, there is a net force acting towards the center which causes the object toseekthe center.To understand the importance of a centripetal force, it is important to have a sturdy understanding of the Newton's first law of motion -

the law of inertia. The law of inertia states that ...- || ... objects in motion tend to stay in motion with the same speed and the same direction unless acted upon by an unbalanced force. ||

According to Newton's first law of motion, it is the natural tendency of all moving objects to continue in motion in the same direction that they are moving ... unless some form of unbalanced force acts upon the object to deviate its motion from its straight-line path. Moving objects will tend to naturally travel in straight lines; an unbalanced force is only required to cause it to turn. Thus, the presence of an unbalanced force is required for objects to move in circles.The Centripetal Force and Direction Change

Any object moving in a circle (or along a circular path) experiences a

centripetal force. That is, there is some physical force pushing or pulling the object towards the center of the circle. This is the centripetal force requirement. The wordcentripetalis merely an adjective used to describe the direction of the force. We are not introducing a newtypeof force but rather describing the direction of the net force acting upon the object that moves in the circle. Whatever the object, if it moves in a circle, there is some force acting upon it to cause it to deviate from its straight-line path, accelerate inwards and move along a circular path. Three such examples of centripetal force are shown below.Newton's Law of Universal GravitationIsaac Newton compared the acceleration of the moon to the acceleration of objects on earth. Believing that gravitational forces were responsible for each, Newton was able to draw an important conclusion about the dependence of gravity upon distance. This comparison led him to conclude that the force of gravitational attraction between the Earth and other objects is inversely proportional to the distance separating the earth's center from the object's center. But distance is not the only variable affecting the magnitude of a gravitational force. Consider Newton's famous equation

Fnet = m • a

Newton knew that the force that caused the apple's acceleration (gravity) must be dependent upon the mass of the apple. And since the force acting to cause the apple's downward acceleration also causes the earth's upward acceleration (Newton's third law), that force must also depend upon the mass of the earth. So for Newton, the force of gravity acting between the earth and any other object is directly proportional to the mass of the earth, directly proportional to the mass of the object, and inversely proportional to the square of the distance that separates the centers of the earth and the object.

The constant of proportionality (G) in the above equation is known as the

universal gravitation constant. The precise value of G was determined experimentally by Henry Cavendish in the century after Newton's death. The value of G is found to beG = 6.673 x 10-11 N m2/kg2The units on G may seem rather odd; nonetheless they are sensible. When the units on G are substituted into the equation above and multiplied by

m1• m2units and divided byd2units, the result will be Newtons - the unit of force.Knowing the value of G allows us to calculate the force of gravitational attraction between any two objects of known mass and known separation distance.

Forces in a Collision

Collisions

Types of Collisions

Perfectly Inelastic Collisions

Kinetic Energy in Inelastic Collisions

Elastic Collisions

Inelastic CollisionCollisions between objects are governed by laws of momentum and energy. When a collision occurs in an isolated system, the total momentum of the system of objects is conserved. Provided that there are no net external forces acting upon the objects, the momentum of all objects before the collision equals the momentum of all objects after the collision. If there are only two objects involved in the collision, then the momentum change of the individual objects are equal in magnitude and opposite in direction.

Certain collisions are referred to as elastic collisions. Elastic collisions are collisions in which both momentum and kinetic energy are conserved. The total system kinetic energy before the collision equals the total system kinetic energy after the collision. If total kinetic energy is not conserved, then the collision is referred to as an inelastic collision.

The animation below portrays the inelastic collision between a 1000-kg car and a 3000-kg truck. The before- and after-collision velocities and momentum are shown in the data tables.

In the collision between the truck and the car, total system momentum is conserved. Before the collision, the momentum of the car is +20000 kg*m/s and the momentum of the truck is -60000 kg*m/s; the total system momentum is -40000 kg*m/s. After the collision, the momentum of the car is -10000 kg*m/s and the momentum of the truck is -30 000 kg*m/s; the total system momentum is -40000 kg*m/s. The total system momentum is conserved. The momentum change of the car (-30000 kg*m/s) is equal in magnitude and opposite in direction to the momentum change of the truck (+30000 kg*m/s) .

An analysis of the kinetic energy of the two objects reveals that the total system kinetic energy before the collision is 800000 Joules (200000 J for the car plus 600000 J for the truck). After the collision, the total system kinetic energy is 200000 Joules (50000 J for the car and 150000 J for the truck). The total kinetic energy before the collision is

notequal to the total kinetic energy after the collision. A large portion of the kinetic energy is converted to other forms of energy such as sound energy and thermal energy. A collision in which total system kinetic energy isnotconserved is known as an inelastic collision.Elastic CollisionCollisions between objects are governed by laws of momentum and energy. When a collision occurs in an isolated system, the total momentum of the system of objects is conserved. Provided that there are no net external forces acting upon the objects, the momentum of all objects before the collision equals the momentum of all objects after the collision. If there are only two objects involved in the collision, then the momentum change of the individual objects are equal in magnitude and opposite in direction.

Certain collisions are referred to as elastic collisions. Elastic collisions are collisions in which both momentum and kinetic energy are conserved. The total system kinetic energy before the collision equals the total system kinetic energy after the collision. If total kinetic energy is not conserved, then the collision is referred to as an inelastic collision.

The animation below portrays the elastic collision between a 1000-kg car and a 3000-kg truck. The before- and after-collision velocities and momentum are shown in the data tables.

In the collision between the truck and the car, total system momentum is conserved. Before the collision, the momentum of the car is +20000 kg*m/s and the momentum of the truck is -60000 kg*m/s; the total system momentum is -40000 kg*m/s. After the collision, the momentum of the car is -40000 kg*m/s and the momentum of the truck is 0 kg*m/s; the total system momentum is -40000 kg*m/s. The total system momentum is conserved. The momentum change of the car (-40000 kg*m/s) is equal in magnitude and opposite in direction to the momentum change of the truck (40000 kg*m/s) .

An analysis of the kinetic energy of the two objects reveals that the total system kinetic energy before the collision is 800000 Joules (200000 J for the car plus 600000 J for the truck). After the collision, the total system kinetic energy is 800000 Joules (800000 J for the car and 0 J for the truck). The total kinetic energy before the collision is equal to the total kinetic energy after the collision. A collision such as this in which total system kinetic energy is conserved is known as an elastic collision.

Momentum

Impulse

Impulse-Momentum Theorem

Law of Conservation of Momentum

Momentum- The sports announcer says, "Going into the all-star break, the Chicago White Sox have the momentum." The headlines declare "Chicago Bulls Gaining Momentum." The coach pumps up his team at half-time, saying "You have the momentum; the critical need is that you use thatmomentum and bury them in this third quarter."

Momentum is a commonly used term in sports. A team that has the momentum is on the move and is going to take some effort to stop. A team that has a lot of momentum is really on the move and is going to be hard to stop. Momentum is a physics term; it refers to the quantity of motion that an object has. A sports team that is on the movehas the momentum. If an object is in motion (on the move) then it has momentum.Momentum can be defined as "mass in motion." All objects have mass; so if an object is moving, then it has momentum - it has its mass in motion. The amount of momentum that an object has is dependent upon two variables: how much stuff is moving and how fast the stuff is moving. Momentum depends upon the variables mass and velocity.In terms of an equation, the momentum of an object is equal to the mass of the object times the velocity of the object.

Momentum = mass • velocity

In physics, the symbol for the quantity momentum is the lower case "p". Thus, the above equation can be rewritten as

p = m • v

where m is the mass and v is the velocity. The equation illustrates that momentum is directly proportional to an object's mass and directly proportional to the object's velocity.

The units for momentum would be mass units times velocity units. The standard metric unit of momentum is the kg•m/s. While the kg•m/s is the standard metric unit of momentum, there are a variety of other units that are acceptable (though not conventional) units of momentum. Examples include kg•mi/hr, kg•km/hr, and g•cm/s. In each of these examples, a mass unit is multiplied by a velocity unit to provide a momentum unit. This is consistent with the equation for momentum.

Momentum is a vector quantity. A vector quantity is a quantity that is fully described by both magnitude and direction. To fully describe the momentum of a 5-kg bowling ball moving westward at 2 m/s, youmust include information about both the magnitude and the direction of the bowling ball. It is not enough to say that the ball has 10 kg•m/s of momentum; the momentum of the ball is not fully described until information about its direction is given. The direction of the momentum vector is the same as the direction of the velocity of the ball. In a previous unit, it was said that the direction of the velocity vector is the same as the direction that an object is moving. If the bowling ball is moving westward, then its momentum can be fully described by saying that it is 10 kg•m/s, westward. As a vector quantity, the momentum of an object is fully described by both magnitude and direction.

From the definition of momentum, it becomes obvious that an object has a large momentum if either its mass or its velocity is large. Both variables are of equal importance in determining the momentum of an object. Consider a Mack truck and a roller skate moving down the street at the same speed. The considerably greater mass of the Mack truck gives it a considerably greater momentum. Yet if the Mack truck were at rest, then the momentum of the least massive roller skate would be the greatest. The momentum of any object that is at rest is 0. Objects at rest do not have momentum - they do not have any "mass in motion." Both variables - mass and velocity - are important in comparing the momentum of two objects.

The momentum equation can help us to think about how a change in one of the two variables might affect the momentum of an object. Consider a 0.5-kg physics cart loaded with one 0.5-kg brick and moving with a speed of 2.0 m/s. The total mass of loaded cart is 1.0 kg and its momentum is 2.0 kg•m/s. If the cart was instead loaded with three 0.5-kg bricks, then the total mass of the loaded cart would be 2.0 kg and its momentum would be 4.0 kg•m/s. A doubling of the mass results in a doubling of the momentum.

Similarly, if the 2.0-kg cart had a velocity of 8.0 m/s (instead of 2.0 m/s), then the cart would have a momentum of 16.0 kg•m/s (instead of 4.0 kg•m/s). A quadrupling in velocity results in a quadrupling of the momentum. These two examples illustrate how the equation p = m•v serves as a "guide to thinking" and not merely a "plug-and-chug recipe for algebraic problem-solving."

Momentum and Impulse ConnectionWhen a sports announcer says that a team has the momentum they mean that the team is really on the move and is going to be hard to stop. The termmomentum is a physics concept. Any object with momentum is going to be hard to stop. To stop such an object, it is necessary to apply a force against its motion for a given period of time. The more momentum that an object has, the harder that it is to stop. Thus, it would require a greater amount of force or a longer amount of time or both to bring such an object to a halt. As the force acts upon the object for a given amount of time, the object's velocity is changed; and hence, the object's momentum is changed.

The concepts in the above paragraph should not seem like abstract information to you. You have observed this a number of times if you have watched the sport of football. In football, the defensive players apply a force for a given amount of time to stop the momentum of the offensive player who has the ball. You have also experienced this a multitude of times while driving. As you bring your car to a halt when approaching a stop sign or stoplight, the brakes serve to apply a force to the car for a given amount of time to change the car's momentum. An object with momentum can be stopped if a force is applied against it for a given amount of time.

A force acting for a given amount of time will change an object's momentum. Put another way, an unbalanced force always accelerates an object - either speeding it up or slowing it down. If the force acts opposite the object's motion, it slows the object down. If a force acts in the same direction as the object's motion, then the force speeds the object up. Either way, a force will change the velocity of an object. And if the velocity of the object is changed, then the momentum of the object is changed.

These concepts are merely an outgrowth of Newton's second law as discussed in an earlier unit. Newton's second law (Fnet = m • a) stated that the acceleration of an object is directly proportional to the net force acting upon the object and inversely proportional to the mass of the object. When combined with the definition of acceleration (a = change in velocity / time), the following equalities result.

If both sides of the above equation are multiplied by the quantity t, a new equation results.

This equation represents one of two primary principles to be used in the analysis of collisions during this unit. To truly understand the equation, it is important to understand its meaning in words. In words, it could be said that the force times the time equals the mass times the change in velocity. In physics, the quantity Force • time is known asimpulse. And since the quantity m•v is the momentum, the quantity m•Δv must be the change in momentum. The equation really says that the

Impulse = Change in momentum

One focus of this unit is to understand the physics of collisions. The physics of collisions are governed by the laws of momentum; and the first law that we discuss in this unit is expressed in the above equation. The equation is known as the impulse-momentum change equation. The law can be expressed this way:

The Law of Action-Reaction (Revisited)

A collision is an interaction between two objects that have made contact (usually) with each other. As in any interaction, a collision results in a force being applied to the two colliding objects. Newton's laws of motion govern such collisions. Newton's third law of motion was introduced and discussed. It was said that...

Newton's third law of motion is naturally applied to collisions between two objects. In a collision between two objects, both objects experience forces that are equal in magnitude and opposite in direction. Such forces often cause one object to speed up (gain momentum) and the other object to slow down (lose momentum). According to Newton's third law, the forces on the two objects are equal in magnitude. While the forces are equal in magnitude and opposite in direction, the accelerations of the objects are not necessarily equal in magnitude. In accord with Newton's second law of motion, the acceleration of an object is dependent upon both force and mass. Thus, if the colliding objects have unequal mass, they will have unequal accelerations as a result of the contact force that results during the collision.

Consider the collision between the club head and the golf ball in the sport of golf. When the club head of a moving golf club collides with a golf ball at rest upon a tee, the force experienced by the club head is equal to the force experienced by the golf ball. Most observers of this collision have difficulty with this concept because they perceive the high speed given to the ball as the result of the collision. They are not observing unequal forces upon the ball and club head, but rather unequal accelerations. Both club head and ball experience equal forces, yet the ball experiences a greater acceleration due to its smaller mass. In a collision, there is a force on both objects that causes an acceleration of both objects. The forces are equal in magnitude and opposite in direction, yet the least massive object receives the greatest acceleration.

Consider the collision between a moving seven ball and an eight ball that is at rest in the sport of table pool. When the seven ball collides with the eight ball, each ball experiences an equal force directed in opposite directions. The rightward moving seven ball experiences a leftward force that causes it to slow down; the eight ball experiences a rightward force that causes it to speed up. Since the two balls have equal masses, they will also experience equal accelerations. In a collision, there is a force on both objects that causes an acceleration of both objects; the forces are equal in magnitude and opposite in direction. For collisions between equal-mass objects, each object experiences the same acceleration.

Consider the interaction between a male and female figure skater in pair figure skating. A woman (m = 45 kg) is kneeling on the shoulders of a man (m = 70 kg); the pair is moving along the ice at 1.5 m/s. The man gracefully tosses the woman forward through the air and onto the ice. The woman receives the forward force and the man receives a backward force. The force on the man is equal in magnitude and opposite in direction to the force on the woman. Yet the acceleration of the woman is greater than the acceleration of the man due to the smaller mass of the woman.

Many observers of this interaction have difficulty believing that the man experienced a backward force. "After all," they might argue, "the man did not move backward." Such observers are presuming that forces cause motion. In their minds, a backward force on the male skater would cause a backward motion. This is a common misconception that has been addressed elsewhere in The Physics Classroom. Forces cause acceleration, not motion. The male figure skater experiences a backwards force that causes his backwards acceleration. The male skater slows down while the woman skater speeds up. In every interaction (with no exception), there are forces acting upon the two interacting objects that are equal in magnitude and opposite in direction.

Collisions are governed by Newton's laws. The law of action-reaction (Newton's third law) explains the nature of the forces between the two interacting objects. According to the law, the force exerted by object 1 upon object 2 is equal in magnitude and opposite in direction to the force exerted by object 2 upon object 1.

WorkEnergyWork- Kinetic Energy TheoremPotential EnergyConservation of EnergyPower

Definition and Mathematics of Work

When a force acts upon an object to cause a displacement of the object, it is said that work was done upon the object. There are three key ingredients to work - force, displacement, and cause. In order for a force to qualify as having done work on an object, there must be a displacement and the force must cause the displacement. There are several good examples of work that can be observed in everyday life - a horse pulling a plow through the field, a father pushing a grocery cart down the aisle of a grocery store, a freshman lifting a backpack full of books upon her shoulder, a weightlifter lifting a barbell above his head, an Olympian launching the shot-put, etc. In each case described here there is a force exerted upon an object to cause that object to be displaced.

Mathematically, work can be expressed by the following equation.

where F is the force, d is the displacement, and the angle (theta) is defined as the angle between the force and the displacement vector. Perhaps the most difficult aspect of the above equation is the angle "theta." The angle is not justany 'ole angle, but rather a very specific angle. The angle measure is defined as the angle between the force and the displacement.

The equation for work lists three variables - each variable is associated with one of the three key words mentioned in the definition of work (force, displacement, and cause). The angle theta in the equation is associated with the amount of force that causes a displacement. When a force is exerted on an object at an angle to the horizontal, only a part of the force contributes to (or causes) a horizontal displacement. Let's consider the force of a chain pulling upwards and rightwards upon Fido in order to drag Fido to the right. It is only the horizontal component of the tension force in the chain that causes Fido to be displaced to the right. The horizontal component is found by multiplying the force F by the cosine of the angle between F and d. In this sense, the cosine theta in the work equation relates to the cause factor - it selects the portion of the force that actually causes a displacement.

Units of Work

Whenever a new quantity is introduced in physics, the standard metric units associated with that quantity are discussed. In the case of work (and also energy), the standard metric unit is the Joule (abbreviated J). One Joule is equivalent to one Newton of force causing a displacement of one meter. In other words,

The Joule is the unit of work.

1 Joule = 1 Newton * 1 meter

1 J = 1 N * m

In summary, work is done when a force acts upon an object to cause a displacement. Three quantities must be known in order to calculate the amount of work. Those three quantities are force, displacement and the angle between the force and the displacement.

Kinetic Energy

Kinetic energy is the energy of motion. An object that has motion - whether it is vertical or horizontal motion - has kinetic energy. There are many forms of kinetic energy - vibrational (the energy due to vibrational motion), rotational (the energy due to rotational motion), and translational (the energy due to motion from one location to another). To keep matters simple, we will focus upon translational kinetic energy. The amount of translational kinetic energy (from here on, the phrase kinetic energy will refer to translational kinetic energy) that an object has depends upon two variables: the mass (m) of the object and the speed (v) of the object. The following equation is used to represent the kinetic energy (KE) of an object.

where m = mass of object

v = speed of object

This equation reveals that the kinetic energy of an object is directly proportional to the square of its speed. That means that for a twofold increase in speed, the kinetic energy will increase by a factor of four. For a threefold increase in speed, the kinetic energy will increase by a factor of nine. And for a fourfold increase in speed, the kinetic energy will increase by a factor of sixteen. The kinetic energy is dependent upon the square of the speed. As it is often said, an equation is not merely a recipe for algebraic problem solving, but also a guide to thinking about the relationship between quantities.

Kinetic energy is a scalar quantity; it does not have a direction. Unlike velocity, acceleration, force, and momentum, the kinetic energy of an object is completely described by magnitude alone. Like work and potential energy, the standard metric unit of measurement for kinetic energy is the Joule. As might be implied by the above equation, 1 Joule is equivalent to 1 kg*(m/s)^2.

Potential Energy

An object can store energy as the result of its position. For example, the heavy ball of a demolition machine is storing energy when it is held at an elevated position. This stored energy of position is referred to as potential energy. Similarly, a drawn bow is able to store energy as the result of its position. When assuming its usual position (i.e., when not drawn), there is no energy stored in the bow. Yet when its position is altered from its usual equilibrium position, the bow is able to store energy by virtue of its position. This stored energy of position is referred to as potential energy. Potential energy is the stored energy of position possessed by an object.

Gravitational Potential Energy

Gravitational potential energy is the energy stored in an object as the result of its vertical position or height. The energy is stored as the result of the gravitational attraction of the Earth for the object. The gravitational potential energy of the massive ball of a demolition machine is dependent on two variables - the mass of the ball and the height to which it is raised. There is a direct relation between gravitational potential energy and the mass of an object. More massive objects have greater gravitational potential energy. There is also a direct relation between gravitational potential energy and the height of an object. The higher that an object is elevated, the greater the gravitational potential energy. These relationships are expressed by the following equation:

In the above equation,

mrepresents the mass of the object,hrepresents the height of the object andgrepresents the gravitational field strength (9.8 N/kg on Earth) - sometimes referred to as the acceleration of gravity. Since the gravitational potential energy of an object is directly proportional to its height above the zero position, adoublingof the height will result in adoublingof the gravitational potential energy. Atriplingof the height will result in atriplingof the gravitational potential energy.Elastic Potential Energy

The second form of potential energy that we will discuss is elastic potential energy.

Elastic potential energyis the energy stored in elastic materials as the result of their stretching or compressing. Elastic potential energy can be stored in rubber bands, bungee chords, trampolines, springs, an arrow drawn into a bow, etc. The amount of elastic potential energy stored in such a device is related to the amount of stretch of the device - the more stretch, the more stored energy.Springs are a special instance of a device that can store elastic potential energy due to either compression or stretching. A force is required to compress a spring; the more compression there is, the more force that is required to compress it further. For certain springs, the amount of force is directly proportional to the amount of stretch or compression (x); the constant of proportionality is known as the spring constant (k).

Such springs are said to follow Hooke's Law. If a spring is not stretched or compressed, then there is no elastic potential energy stored in it. The spring is said to be at its equilibrium position. The equilibrium position is the position that the spring naturally assumes when there is no force applied to it. In terms of potential energy, the equilibrium position could be called the zero-potential energy position. There is a special equation for springs that relates the amount of elastic potential energy to the amount of stretch (or compression) and the spring constant. The equation is

To summarize, potential energy is the energy that is stored in an object due to its position relative to some zero position. An object possesses gravitational potential energy if it is positioned at a height above (or below) the zero height. An object possesses elastic potential energy if it is at a position on an elastic medium other than the equilibrium position.

Mechanical Energy

It was said that work is done upon an object whenever a force acts upon it to cause it to be displaced. Work involves a force acting upon an object to cause a displacement. In all instances in which work is done, there is an object that supplies the force in order to do the work. If a World Civilization book is lifted to the top shelf of a student locker, then the student supplies the force to do the work on the book. If a plow is displaced across a field, then some form of farm equipment (usually a tractor or a horse) supplies the force to do the work on the plow. If a pitcher winds up and accelerates a baseball towards home plate, then the pitcher supplies the force to do the work on the baseball. If a roller coaster car is displaced from ground level to the top of the first drop of a roller coaster ride, then a chain driven by a motor supplies the force to do the work on the car. If a barbell is displaced from ground level to a height above a weightlifter's head, then the weightlifter is supplying a force to do work on the barbell. In all instances, an object that possesses some form of energy supplies the force to do the work. In the instances described here, the objects doing the work (a student, a tractor, a pitcher, a motor/chain) possess chemical potential energy stored in food or fuel that is transformed into work. In the process of doing work, the object that is doing the work exchanges energy with the object upon which the work is done. When the work is done upon the object, that object gains energy. The energy acquired by the objects upon which work is done is known as mechanical energy.

Mechanical energy is the energy that is possessed by an object due to its motion or due to its position. Mechanical energy can be either kinetic energy (energy of motion) or potential energy (stored energy of position). Objects have mechanical energy if they are in motion and/or if they are at some position relative to a zero potential energy position (for example, a brick held at a vertical position above the ground or zero height position). A moving car possesses mechanical energy due to its motion (kinetic energy). A moving baseball possesses mechanical energy due to both its high speed (kinetic energy) and its vertical position above the ground (gravitationalpotential energy). A World Civilization book at rest on the top shelf of a locker possesses mechanical energy due to its vertical position above the ground (gravitational potential energy). A barbell lifted high above a weightlifter's head possesses mechanical energy due to its vertical position above the ground (gravitational potential energy). A drawn bow possesses mechanical energy due to its stretched position (elasticpotential energy).

The Total Mechanical Energy

As already mentioned, the mechanical energy of an object can be the result of its motion (i.e., kinetic energy) and/or the result of its stored energy of position (i.e., potential energy). The total amount of mechanical energy is merely the sum of the potential energy and the kinetic energy. This sum is simply referred to as the total mechanical energy (abbreviated TME).

TME = PE + KE

As discussed earlier, there are two forms of potential energy discussed in our course - gravitational potential energy and elastic potential energy. Given this fact, the above equation can be rewritten:

TME = PEgrav + PEspring + KE

Power

The quantity work has to do with a force causing a displacement. Work has nothing to do with the amount of time that this force acts to cause the displacement. Sometimes, the work is done very quickly and other times the work is done rather slowly. For example, a rock climber takes an abnormally long time to elevate her body up a few meters along the side of a cliff. On the other hand, a trail hiker (who selects the easier path up the mountain) might elevate her body a few meters in a short amount of time. The two people might do the same amount of work, yet the hiker does the work in considerably less time than the rock climber. The quantity that has to do with the rate at which a certain amount of work is done is known as the power. The hiker has a greater power rating than the rock climber.

Power is the rate at which work is done. It is the work/time ratio. Mathematically, it is computed using the following equation.

The standard metric unit of power is the Watt. As is implied by the equation for power, a unit of power is equivalent to a unit of work divided by a unit of time. Thus, a Watt is equivalent to a Joule/second. For historical reasons, the horsepower is occasionally used to describe the power delivered by a machine. One horsepower is equivalent to approximately 750 Watts.

Most machines are designed and built to do work on objects. All machines are typically described by a power rating. The power rating indicates the rate at which that machine can do work upon other objects. Thus, the power of a machine is the work/time ratio for that particular machine. A car engine is an example of a machine that is given a power rating. The power rating relates to how rapidly the car can accelerate the car. Suppose that a 40-horsepower engine could accelerate the car from 0 mi/hr to 60 mi/hr in 16 seconds. If this were the case, then a car with four times the horsepower could do the same amount of work in one-fourth the time. That is, a 160-horsepower engine could accelerate the same car from 0 mi/hr to 60 mi/hr in 4 seconds. The point is that for the same amount of work, power and time are inversely proportional. The power equation suggests that a more powerful engine can do the same amount of work in less time.

A person is also a machine that has a power rating. Some people are more power-full than others. That is, some people are capable of doing the same amount of work in less time or more work in the same amount of time. A common physics lab involves quickly climbing a flight of stairs and using mass, height and time information to determine a student's personal power. Despite the diagonal motion along the staircase, it is often assumed that the horizontal motion is constant and all the force from the steps is used to elevate the student upward at a constant speed. Thus, the weight of the student is equal to the force that does the work on the student and the height of the staircase is the upward displacement.