# Uniform Circular Motion and Gravitation

## Satellites and Kepler’s Laws: An Argument for Simplicity

### Learning Objectives

By the end of this section, you will be able to:
1. State Kepler’s laws of planetary motion.
2. Derive the third Kepler’s law for circular orbits.
3. Discuss the Ptolemaic model of the universe.
Examples of gravitational orbits abound. Hundreds of artificial satellites orbit Earth together with thousands of pieces of debris. The Moon’s orbit about Earth has intrigued humans from time immemorial. The orbits of planets, asteroids, meteors, and comets about the Sun are no less interesting. If we look further, we see almost unimaginable numbers of stars, galaxies, and other celestial objects orbiting one another and interacting through gravity.
All these motions are governed by gravitational force, and it is possible to describe them to various degrees of precision. Precise descriptions of complex systems must be made with large computers. However, we can describe an important class of orbits without the use of computers, and we shall find it instructive to study them. These orbits have the following characteristics:
1. A small mass . This allows us to view the motion as if  were stationary—in fact, as if from an inertial frame of reference placed on  —without significant error. Mass  is the satellite of , if the orbit is gravitationally bound.
2. The system is isolated from other masses. This allows us to neglect any small effects due to outside masses.
The conditions are satisfied, to good approximation, by Earth’s satellites (including the Moon), by objects orbiting the Sun, and by the satellites of other planets. Historically, planets were studied first, and there is a classical set of three laws, called Kepler’s laws of planetary motion, that describe the orbits of all bodies satisfying the two previous conditions (not just planets in our solar system). These descriptive laws are named for the German astronomer Johannes Kepler (1571–1630), who devised them after careful study (over some 20 years) of a large amount of meticulously recorded observations of planetary motion done by Tycho Brahe (1546–1601). Such careful collection and detailed recording of methods and data are hallmarks of good science. Data constitute the evidence from which new interpretations and meanings can be constructed.

### Kepler’s Laws of Planetary Motion

      Kepler’s First Law

The orbit of each planet about the Sun is an ellipse with the Sun at one focus.

      Kepler’s Second Law

Each planet moves so that an imaginary line drawn from the Sun to the planet sweeps out equal areas in equal times (see ).

      Kepler’s Third Law

The ratio of the squares of the periods of any two planets about the Sun is equal to the ratio of the cubes of their average distances from the Sun. In equation form, this is
where  is the period (time for one orbit) and  is the average radius. This equation is valid only for comparing two small masses orbiting the same large one. Most importantly, this is a descriptive equation only, giving no information as to the cause of the equality.
Note again that while, for historical reasons, Kepler’s laws are stated for planets orbiting the Sun, they are actually valid for all bodies satisfying the two previously stated conditions.
People immediately search for deeper meaning when broadly applicable laws, like Kepler’s, are discovered. It was Newton who took the next giant step when he proposed the law of universal gravitation. While Kepler was able to discover what was happening, Newton discovered that gravitational force was the cause.

### Derivation of Kepler’s Third Law for Circular Orbits

We shall derive Kepler’s third law, starting with Newton’s laws of motion and his universal law of gravitation. The point is to demonstrate that the force of gravity is the cause for Kepler’s laws (although we will only derive the third one).
Let us consider a circular orbit of a small mass  around a large mass , satisfying the two conditions stated at the beginning of this section. Gravity supplies the centripetal force to mass . Starting with Newton’s second law applied to circular motion,
The net external force on mass  is gravity, and so we substitute the force of gravity for :
The mass  cancels, yielding
The fact that  cancels out is another aspect of the oft-noted fact that at a given location all masses fall with the same acceleration. Here we see that at a given orbital radius , all masses orbit at the same speed. (This was implied by the result of the preceding worked example.) Now, to get at Kepler’s third law, we must get the period  into the equation. By definition, period  is the time for one complete orbit. Now the average speed  is the circumference divided by the period—that is,
Substituting this into the previous equation gives
Solving for  yields
Using subscripts 1 and 2 to denote two different satellites, and taking the ratio of the last equation for satellite 1 to satellite 2 yields
This is Kepler’s third law. Note that Kepler’s third law is valid only for comparing satellites of the same parent body, because only then does the mass of the parent body  cancel.
Now consider what we get if we solve  for the ratio . We obtain a relationship that can be used to determine the mass  of a parent body from the orbits of its satellites:
If  and  are known for a satellite, then the mass  of the parent can be calculated. This principle has been used extensively to find the masses of heavenly bodies that have satellites. Furthermore, the ratio  should be a constant for all satellites of the same parent body (because ). (See ).
It is clear from  that the ratio of  is constant, at least to the third digit, for all listed satellites of the Sun, and for those of Jupiter. Small variations in that ratio have two causes—uncertainties in the  and  data, and perturbations of the orbits due to other bodies. Interestingly, those perturbations can be—and have been—used to predict the location of new planets and moons. This is another verification of Newton’s universal law of gravitation.

### The Case for Simplicity

The development of the universal law of gravitation by Newton played a pivotal role in the history of ideas. While it is beyond the scope of this text to cover that history in any detail, we note some important points. The definition of planet set in 2006 by the International Astronomical Union (IAU) states that in the solar system, a planet is a celestial body that:
1. is in orbit around the Sun,
2. has sufficient mass to assume hydrostatic equilibrium and
3. has cleared the neighborhood around its orbit.
A non-satellite body fulfilling only the first two of the above criteria is classified as “dwarf planet.”
In 2006, Pluto was demoted to a ‘dwarf planet’ after scientists revised their definition of what constitutes a “true” planet.
The universal law of gravitation is a good example of a physical principle that is very broadly applicable. That single equation for the gravitational force describes all situations in which gravity acts. It gives a cause for a vast number of effects, such as the orbits of the planets and moons in the solar system. It epitomizes the underlying unity and simplicity of physics.
Before the discoveries of Kepler, Copernicus, Galileo, Newton, and others, the solar system was thought to revolve around Earth as shown in (a). This is called the Ptolemaic view, for the Greek philosopher who lived in the second century AD. This model is characterized by a list of facts for the motions of planets with no cause and effect explanation. There tended to be a different rule for each heavenly body and a general lack of simplicity.
(b) represents the modern or Copernican model. In this model, a small set of rules and a single underlying force explain not only all motions in the solar system, but all other situations involving gravity. The breadth and simplicity of the laws of physics are compelling. As our knowledge of nature has grown, the basic simplicity of its laws has become ever more evident.

### Section Summary

1. Kepler’s laws are stated for a small mass  orbiting a larger mass  in near-isolation. Kepler’s laws of planetary motion are then as follows:
    Kepler’s first law
    The orbit of each planet about the Sun is an ellipse with the Sun at one focus.
    Kepler’s second law
    Each planet moves so that an imaginary line drawn from the Sun to the planet sweeps out equal areas in equal times.
    Kepler’s third law
    The ratio of the squares of the periods of any two planets about the Sun is equal to the ratio of the cubes of their average distances from the Sun:
    where
2. The period and radius of a satellite’s orbit about a larger body  are related by

    or

### Conceptual Questions

### Problem Exercises


# Work, Energy, and Energy Resources

## Connection for AP® Courses

Energy plays an essential role both in everyday events and in scientific phenomena. You can no doubt name many forms of energy, from that provided by our foods to the energy we use to run our cars and the sunlight that warms us on the beach. You can also cite examples of what people call “energy” that may not be scientific, such as someone having an energetic personality. Not only does energy have many interesting forms, it is involved in almost all phenomena, and is one of the most important concepts of physics.
There is no simple and accurate scientific definition for energy. Energy is characterized by its many forms and the fact that it is conserved. We can loosely define energy as the ability to do work, admitting that in some circumstances not all energy is available to do work. Because of the association of energy with work, we begin the chapter with a discussion of work. Work is intimately related to energy and how energy moves from one system to another or changes form. The work-energy theorem supports Big Idea 3, that interactions between objects are described by forces. In particular, exerting a force on an object may do work on it, changing it's energy (Enduring Understanding 3.E). The work-energy theorem, introduced in this chapter, establishes the relationship between work done on an object by an external force and changes in the object’s kinetic energy (Essential Knowledge 3.E.1).
Similarly, systems can do work on each other, supporting Big Idea 4, that interactions between systems can result in changes in those systems—in this case, changes in the total energy of the system (Enduring Understanding 4.C). The total energy of the system is the sum of its kinetic energy, potential energy, and microscopic internal energy (Essential Knowledge 4.C.1). In this chapter students learn how to calculate kinetic, gravitational, and elastic potential energy in order to determine the total mechanical energy of a system. The transfer of mechanical energy into or out of a system is equal to the work done on the system by an external force with a nonzero component parallel to the displacement (Essential Knowledge 4.C.2).
An important aspect of energy is that the total amount of energy in the universe is constant. Energy can change forms, but it cannot appear from nothing or disappear without a trace. Energy is thus one of a handful of physical quantities that we say is “conserved.” Conservation of energy (as physicists call the principle that energy can neither be created nor destroyed) is based on experiment. Even as scientists discovered new forms of energy, conservation of energy has always been found to apply. Perhaps the most dramatic example of this was supplied by Einstein when he suggested that mass is equivalent to energy (his famous equation E = mc2). This is one of the most important applications of Big Idea 5, that changes that occur as a result of interactions are constrained by conservation laws. Specifically, there are many situations where conservation of energy (Enduring Understanding 5.B) is both a useful concept and starting point for calculations related to the system. Note, however, that conservation doesn’t necessarily mean that energy in a system doesn’t change. Energy may be transferred into or out of the system, and the change must be equal to the amount transferred (Enduring Understanding 5.A). This may occur if there is an external force or a transfer between external objects and the system (Essential Knowledge 5.A.3). Energy is one of the fundamental quantities that are conserved for all systems (Essential Knowledge 5.A.2). The chapter introduces concepts of kinetic energy and potential energy. Kinetic energy is introduced as an energy of motion that can be changed by the amount of work done by an external force. Potential energy can only exist when objects interact with each other via conservative forces according to classical physics (Essential Knowledge 5.B.3). Because of this, a single object can only have kinetic energy and no potential energy (Essential Knowledge 5.B.1). The chapter also introduces the idea that the energy transfer is equal to the work done on the system by external forces and the rate of energy transfer is defined as power (Essential Knowledge 5.B.5).
From a societal viewpoint, energy is one of the major building blocks of modern civilization. Energy resources are key limiting factors to economic growth. The world use of energy resources, especially oil, continues to grow, with ominous consequences economically, socially, politically, and environmentally. We will briefly examine the world’s energy use patterns at the end of this chapter.
The concepts in this chapter support:
Big Idea 3 The interactions of an object with other objects can be described by forces.
Enduring Understanding 3.E A force exerted on an object can change the kinetic energy of the object.
Essential Knowledge 3.E.1 The change in the kinetic energy of an object depends on the force exerted on the object and on the displacement of the object during the interval that the force is exerted.
Big Idea 4 Interactions between systems can result in changes in those systems.
Enduring Understanding 4.C Interactions with other objects or systems can change the total energy of a system.
Essential Knowledge 4.C.1 The energy of a system includes its kinetic energy, potential energy, and microscopic internal energy. Examples should include gravitational potential energy, elastic potential energy, and kinetic energy.
Essential Knowledge 4.C.2 Mechanical energy (the sum of kinetic and potential energy) is transferred into or out of a system when an external force is exerted on a system such that a component of the force is parallel to its displacement. The process through which the energy is transferred is called work.
Big Idea 5 Changes that occur as a result of interactions are constrained by conservation laws.
Enduring Understanding 5.A Certain quantities are conserved, in the sense that the changes of those quantities in a given system are always equal to the transfer of that quantity to or from the system by all possible interactions with other systems.
Essential Knowledge 5.A.2 For all systems under all circumstances, energy, charge, linear momentum, and angular momentum are conserved.
Essential Knowledge 5.A.3 An interaction can be either a force exerted by objects outside the system or the transfer of some quantity with objects outside the system.
Enduring Understanding 5.B The energy of a system is conserved.
Essential Knowledge 5.B.1 Classically, an object can only have kinetic energy since potential energy requires an interaction between two or more objects.
Essential Knowledge 5.B.3 A system with internal structure can have potential energy. Potential energy exists within a system if the objects within that system interact with conservative forces.
Essential Knowledge 5.B.5 Energy can be transferred by an external force exerted on an object or system that moves the object or system through a distance; this energy transfer is called work. Energy transfer in mechanical or electrical systems may occur at different rates. Power is deﬁned as the rate of energy transfer into, out of, or within a system.


# Work, Energy, and Energy Resources

## Work: The Scientific Definition

### Learning Objectives

By the end of this section, you will be able to:
1. Explain how an object must be displaced for a force on it to do work.
2. Explain how relative directions of force and displacement determine whether the work done is positive, negative, or zero.

### What It Means to Do Work

The scientific definition of work differs in some ways from its everyday meaning. Certain things we think of as hard work, such as writing an exam or carrying a heavy load on level ground, are not work as defined by a scientist. The scientific definition of work reveals its relationship to energy—whenever work is done, energy is transferred.
For work, in the scientific sense, to be done, a force must be exerted and there must be displacement in the direction of the force.
Formally, the work done on a system by a constant force is defined to be the product of the component of the force in the direction of motion times the distance through which the force acts. For one-way motion in one dimension, this is expressed in equation form as
where  is work,  is the displacement of the system, and  is the angle between the force vector  and the displacement vector , as in . We can also write this as
To find the work done on a system that undergoes motion that is not one-way or that is in two or three dimensions, we divide the motion into one-way one-dimensional segments and add up the work done over each segment.
To examine what the definition of work means, let us consider the other situations shown in . The person holding the briefcase in (b) does no work, for example. Here , so . Why is it you get tired just holding a load? The answer is that your muscles are doing work against one another, but they are doing no work on the system of interest (the “briefcase-Earth system”—see Gravitational Potential Energy for more details). There must be displacement for work to be done, and there must be a component of the force in the direction of the motion. For example, the person carrying the briefcase on level ground in (c) does no work on it, because the force is perpendicular to the motion. That is, , and so .
In contrast, when a force exerted on the system has a component in the direction of motion, such as in (d), work is done—energy is transferred to the briefcase. Finally, in (e), energy is transferred from the briefcase to a generator. There are two good ways to interpret this energy transfer. One interpretation is that the briefcase’s weight does work on the generator, giving it energy. The other interpretation is that the generator does negative work on the briefcase, thus removing energy from it. The drawing shows the latter, with the force from the generator upward on the briefcase, and the displacement downward. This makes , and ; therefore,  is negative.

### Calculating Work

Work and energy have the same units. From the definition of work, we see that those units are force times distance. Thus, in SI units, work and energy are measured in newton-meters. A newton-meter is given the special name joule (J), and . One joule is not a large amount of energy; it would lift a small 100-gram apple a distance of about 1 meter.

### Test Prep for AP Courses

### Section Summary

1. Work is the transfer of energy by a force acting on an object as it is displaced.
2. The work  that a force  does on an object is the product of the magnitude  of the force, times the magnitude  of the displacement, times the cosine of the angle  between them. In symbols,

3. The SI unit for work and energy is the joule (J), where .
4. The work done by a force is zero if the displacement is either zero or perpendicular to the force.
5. The work done is positive if the force and displacement have the same direction, and negative if they have opposite direction.

### Conceptual Questions

### Problems & Exercises


# Work, Energy, and Energy Resources

## Kinetic Energy and the Work-Energy Theorem

### Learning Objectives

By the end of this section, you will be able to:
1. Explain work as a transfer of energy and net work as the work done by the net force.
2. Explain and apply the work-energy theorem.

### Work Transfers Energy

What happens to the work done on a system? Energy is transferred into the system, but in what form? Does it remain in the system or move on? The answers depend on the situation. For example, if the lawn mower in (a) is pushed just hard enough to keep it going at a constant speed, then energy put into the mower by the person is removed continuously by friction, and eventually leaves the system in the form of heat transfer. In contrast, work done on the briefcase by the person carrying it up stairs in (d) is stored in the briefcase-Earth system and can be recovered at any time, as shown in (e). In fact, the building of the pyramids in ancient Egypt is an example of storing energy in a system by doing work on the system. Some of the energy imparted to the stone blocks in lifting them during construction of the pyramids remains in the stone-Earth system and has the potential to do work.
In this section we begin the study of various types of work and forms of energy. We will find that some types of work leave the energy of a system constant, for example, whereas others change the system in some way, such as making it move. We will also develop definitions of important forms of energy, such as the energy of motion.

### Net Work and the Work-Energy Theorem

We know from the study of Newton’s laws in Dynamics: Force and Newton's Laws of Motion that net force causes acceleration. We will see in this section that work done by the net force gives a system energy of motion, and in the process we will also find an expression for the energy of motion.

Let us start by considering the total, or net, work done on a system. Net work is defined to be the sum of work on an object. The net work can be written in terms of the net force on an object. . In equation form, this is  where  is the angle between the force vector and the displacement vector.
(a) shows a graph of force versus displacement for the component of the force in the direction of the displacement—that is, an  vs.  graph. In this case,  is constant. You can see that the area under the graph is , or the work done. (b) shows a more general process where the force varies. The area under the curve is divided into strips, each having an average force . The work done is  for each strip, and the total work done is the sum of the . Thus the total work done is the total area under the curve, a useful property to which we shall refer later.
Net work will be simpler to examine if we consider a one-dimensional situation where a force is used to accelerate an object in a direction parallel to its initial velocity. Such a situation occurs for the package on the roller belt conveyor system shown in .
The force of gravity and the normal force acting on the package are perpendicular to the displacement and do no work. Moreover, they are also equal in magnitude and opposite in direction so they cancel in calculating the net force. The net force arises solely from the horizontal applied force  and the horizontal friction force . Thus, as expected, the net force is parallel to the displacement, so that  and , and the net work is given by
The effect of the net force  is to accelerate the package from  to . The kinetic energy of the package increases, indicating that the net work done on the system is positive. (See .) By using Newton’s second law, and doing some algebra, we can reach an interesting conclusion. Substituting  from Newton’s second law gives
To get a relationship between net work and the speed given to a system by the net force acting on it, we take  and use the equation studied in Motion Equations for Constant Acceleration in One Dimension for the change in speed over a distance  if the acceleration has the constant value
; namely,
 (note that
 appears in the expression for the net work). Solving for acceleration gives
. When
 is substituted into the preceding expression for
, we obtain
The  cancels, and we rearrange this to obtain
This expression is called the work-energy theorem, and it actually applies in general (even for forces that vary in direction and magnitude), although we have derived it for the special case of a constant force parallel to the displacement. The theorem implies that the net work on a system equals the change in the quantity . This quantity is our first example of a form of energy.

The quantity  in the work-energy theorem is defined to be the translational kinetic energy (KE) of a mass  moving at a speed . (Translational kinetic energy is distinct from rotational kinetic energy, which is considered later.) In equation form, the translational kinetic energy,
is the energy associated with translational motion. Kinetic energy is a form of energy associated with the motion of a particle, single body, or system of objects moving together.
We are aware that it takes energy to get an object, like a car or the package in , up to speed, but it may be a bit surprising that kinetic energy is proportional to speed squared. This proportionality means, for example, that a car traveling at 100 km/h has four times the kinetic energy it has at 50 km/h, helping to explain why high-speed collisions are so devastating. We will now consider a series of examples to illustrate various aspects of work and energy.
Some of the examples in this section can be solved without considering energy, but at the expense of missing out on gaining insights about what work and energy are doing in this situation. On the whole, solutions involving energy are generally shorter and easier than those using kinematics and dynamics alone.

### Test Prep for AP Courses

### Section Summary

1. The net work  is the work done by the net force acting on an object.
2. Work done on an object transfers energy to the object.
3. The translational kinetic energy of an object of mass  moving at speed  is .
4. The work-energy theorem states that the net work  on a system changes its kinetic energy, .

### Conceptual Questions

### Problems & Exercises


# Work, Energy, and Energy Resources

## Gravitational Potential Energy

### Learning Objectives

By the end of this section, you will be able to:
1. Explain gravitational potential energy in terms of work done against gravity.
2. Show that the gravitational potential energy of an object of mass  at height  on Earth is given by .
3. Show how knowledge of the potential energy as a function of position can be used to simplify calculations and explain physical phenomena.

### Work Done Against Gravity

Climbing stairs and lifting objects is work in both the scientific and everyday sense—it is work done against the gravitational force. When there is work, there is a transformation of energy. The work done against the gravitational force goes into an important form of stored energy that we will explore in this section.
Let us calculate the work done in lifting an object of mass  through a height , such as in , near Earth's surface. If the object is lifted straight up at constant speed, then the force needed to lift it is equal to its weight . The work done on the mass is then . We define this to be the gravitational potential energy  put into (or gained by) the object-Earth system. This energy is associated with the state of separation between two objects that attract each other by the gravitational force. For convenience, we refer to this as the  gained by the object, recognizing that this is energy stored in the gravitational field of Earth. Why do we use the word “system”? Potential energy is a property of a system rather than of a single object—due to its physical position. An object’s gravitational potential is due to its position relative to the surroundings within the Earth-object system. The force applied to the object is an external force, from outside the system. When it does positive work it increases the gravitational potential energy of the system. Because gravitational potential energy depends on relative position, we need a reference level at which to set the potential energy equal to 0. We usually choose this point to be Earth’s surface, but this point is arbitrary; what is important is the difference in gravitational potential energy, because this difference is what relates to the work done. The difference in gravitational potential energy of an object (in the Earth-object system) between two rungs of a ladder will be the same for the first two rungs as for the last two rungs.

### Converting Between Potential Energy and Kinetic Energy

Gravitational potential energy may be converted to other forms of energy, such as kinetic energy. If we release the mass, gravitational force will do an amount of work equal to  on it, thereby increasing its kinetic energy by that same amount (by the work-energy theorem). We will find it more useful to consider just the conversion of  to  without explicitly considering the intermediate step of work. (See .) This shortcut makes it is easier to solve problems using energy (if possible) rather than explicitly using forces.
More precisely, we define the change in gravitational potential energy  to be
where, for simplicity, we denote the change in height by  rather than the usual . Note that  is positive when the final height is greater than the initial height, and vice versa. For example, if a 0.500-kg mass hung from a cuckoo clock is raised 1.00 m, then its change in gravitational potential energy is
Note that the units of gravitational potential energy turn out to be joules, the same as for work and other forms of energy. As the clock runs, the mass is lowered. We can think of the mass as gradually giving up its 4.90 J of gravitational potential energy, without directly considering the force of gravity that does the work.

### Using Potential Energy to Simplify Calculations

The equation  applies for any path that has a change in height of , not just when the mass is lifted straight up, as long as  is small compared to the radius of Earth. Note that, as we learned in Uniform Circular Motion and Gravitation, the force of Earth’s gravity does decrease with distance from Earth. The change is negligible for small changes in distance, but if you want to use potential energy in problems involving travel to the moon or even further, Newton’s universal law of gravity must be taken into account. This more complete treatment is beyond the scope of this text and is not necessary for the problems we consider here. (See .) It is much easier to calculate  (a simple multiplication) than it is to calculate the work done along a complicated path. The idea of gravitational potential energy has the double advantage that it is very broadly applicable and it makes calculations easier. From now on, we will consider that any change in vertical position  of a mass  is accompanied by a change in gravitational potential energy , and we will avoid the equivalent but more difficult task of calculating work done by or against the gravitational force.
We have seen that work done by or against the gravitational force depends only on the starting and ending points, and not on the path between, allowing us to define the simplifying concept of gravitational potential energy. We can do the same thing for a few other forces, and we will see that this leads to a formal definition of the law of conservation of energy.

### Test Prep for AP Courses

### Section Summary

1. Work done against gravity in lifting an object becomes potential energy of the object-Earth system.
2. The change in gravitational potential energy, , is , with  being the increase in height and  the acceleration due to gravity.
3. The gravitational potential energy of an object near Earth’s surface is due to its position in the mass-Earth system. Only differences in gravitational potential energy, , have physical significance.
4. As an object descends without friction, its gravitational potential energy changes into kinetic energy corresponding to increasing speed, so that .

### Conceptual Questions

### Problems & Exercises


# Work, Energy, and Energy Resources

## Conservative Forces and Potential Energy

### Learning Objectives

By the end of this section, you will be able to:
1. Define conservative force, potential energy, and mechanical energy.
2. Explain the potential energy of a spring in terms of its compression when Hooke’s law applies.
3. Use the work-energy theorem to show how having only conservative forces implies conservation of mechanical energy.

### Potential Energy and Conservative Forces

Work is done by a force, and some forces, such as weight, have special characteristics. A conservative force is one, like the gravitational force, for which work done by or against it depends only on the starting and ending points of a motion and not on the path taken. We can define a potential energy  for any conservative force, just as we did for the gravitational force. For example, when you wind up a toy, an egg timer, or an old-fashioned watch, you do work against its spring and store energy in it. (We treat these springs as ideal, in that we assume there is no friction and no production of thermal energy.) This stored energy is recoverable as work, and it is useful to think of it as potential energy contained in the spring. Indeed, the reason that the spring has this characteristic is that its force is conservative. That is, a conservative force results in stored or potential energy. Gravitational potential energy is one example, as is the energy stored in a spring. We will also see how conservative forces are related to the conservation of energy.

### Potential Energy of a Spring

First, let us obtain an expression for the potential energy stored in a spring (). We calculate the work done to stretch or compress a spring that obeys Hooke’s law. (Hooke’s law was examined in Elasticity: Stress and Strain, and states that the magnitude of force  on the spring and the resulting deformation  are proportional, .) (See .) For our spring, we will replace  (the amount of deformation produced by a force ) by the distance  that the spring is stretched or compressed along its length. So the force needed to stretch the spring has magnitude , where  is the spring’s force constant. The force increases linearly from 0 at the start to   in the fully stretched position. The average force is . Thus the work done in stretching or compressing the spring is . Alternatively, we noted in Kinetic Energy and the Work-Energy Theorem that the area under a graph of  vs.  is the work done by the force. In (c) we see that this area is also . We therefore define the potential energy of a spring, , to be
where  is the spring’s force constant and  is the displacement from its undeformed position. The potential energy represents the work done on the spring and the energy stored in it as a result of stretching or compressing it a distance . The potential energy of the spring  does not depend on the path taken; it depends only on the stretch or squeeze  in the final configuration.
The equation  has general validity beyond the special case for which it was derived. Potential energy can be stored in any elastic medium by deforming it. Indeed, the general definition of potential energy is energy due to position, shape, or configuration. For shape or position deformations, stored energy is , where  is the force constant of the particular system and  is its deformation. Another example is seen in  for a guitar string.

### Conservation of Mechanical Energy

Let us now consider what form the work-energy theorem takes when only conservative forces are involved. This will lead us to the conservation of energy principle. The work-energy theorem states that the net work done by all forces acting on a system equals its change in kinetic energy. In equation form, this is
If only conservative forces act, then
where  is the total work done by all conservative forces. Thus,
Now, if the conservative force, such as the gravitational force or a spring force, does work, the system loses potential energy. That is, . Therefore,
or
This equation means that the total kinetic and potential energy is constant for any process involving only conservative forces. That is,
where i and f denote initial and final values. This equation is a form of the work-energy theorem for conservative forces; it is known as the conservation of mechanical energy principle. Remember that this applies to the extent that all the forces are conservative, so that friction is negligible. The total kinetic plus potential energy of a system is defined to be its mechanical energy, . In a system that experiences only conservative forces, there is a potential energy associated with each force, and the energy only changes form between  and the various types of , with the total energy remaining constant.
Note that, for conservative forces, we do not directly calculate the work they do; rather, we consider their effects through their corresponding potential energies, just as we did in . Note also that we do not consider details of the path taken—only the starting and ending points are important (as long as the path is not impossible). This assumption is usually a tremendous simplification, because the path may be complicated and forces may vary along the way.

### Test Prep for AP Courses

### Section Summary

1. A conservative force is one for which work depends only on the starting and ending points of a motion, not on the path taken.
2. We can define potential energy  for any conservative force, just as we defined  for the gravitational force.
3. The potential energy of a spring is , where  is the spring’s force constant and  is the displacement from its undeformed position.
4. Mechanical energy is defined to be  for a conservative force.
5. When only conservative forces act on and within a system, the total mechanical energy is constant. In equation form,
where i and f denote initial and final values. This is known as the conservation of mechanical energy.

### Conceptual Questions

### Problems & Exercises


# Work, Energy, and Energy Resources

## Nonconservative Forces

### Learning Objectives

By the end of this section, you will be able to:
1. Define nonconservative forces and explain how they affect mechanical energy.
2. Show how the principle of conservation of energy can be applied by treating the conservative forces in terms of their potential energies and any nonconservative forces in terms of the work they do.

### Nonconservative Forces and Friction

Forces are either conservative or nonconservative. Conservative forces were discussed in Conservative Forces and Potential Energy. A nonconservative force is one for which work depends on the path taken. Friction is a good example of a nonconservative force. As illustrated in , work done against friction depends on the length of the path between the starting and ending points. Because of this dependence on path, there is no potential energy associated with nonconservative forces. An important characteristic is that the work done by a nonconservative force adds or removes mechanical energy from a system. Friction, for example, creates thermal energy that dissipates, removing energy from the system. Furthermore, even if the thermal energy is retained or captured, it cannot be fully converted back to work, so it is lost or not recoverable in that sense as well.

### How Nonconservative Forces Affect Mechanical Energy

Mechanical energy may not be conserved when nonconservative forces act. For example, when a car is brought to a stop by friction on level ground, it loses kinetic energy, which is dissipated as thermal energy, reducing its mechanical energy.  compares the effects of conservative and nonconservative forces. We often choose to understand simpler systems such as that described in (a) first before studying more complicated systems as in (b).

### How the Work-Energy Theorem Applies

Now let us consider what form the work-energy theorem takes when both conservative and nonconservative forces act. We will see that the work done by nonconservative forces equals the change in the mechanical energy of a system. As noted in  Kinetic Energy and the Work-Energy Theorem, the work-energy theorem states that the net work on a system equals the change in its kinetic energy, or . The net work is the sum of the work by nonconservative forces plus the work by conservative forces. That is,
so that
where  is the total work done by all nonconservative forces and  is the total work done by all conservative forces.
Consider , in which a person pushes a crate up a ramp and is opposed by friction. As in the previous section, we note that work done by a conservative force comes from a loss of gravitational potential energy, so that . Substituting this equation into the previous one and solving for  gives
This equation means that the total mechanical energy  changes by exactly the amount of work done by nonconservative forces. In , this is the work done by the person minus the work done by friction. So even if energy is not conserved for the system of interest (such as the crate), we know that an equal amount of work was done to cause the change in total mechanical energy.
We rearrange  to obtain
This means that the amount of work done by nonconservative forces adds to the mechanical energy of a system. If  is positive, then mechanical energy is increased, such as when the person pushes the crate up the ramp in . If  is negative, then mechanical energy is decreased, such as when the rock hits the ground in (b). If  is zero, then mechanical energy is conserved, and nonconservative forces are balanced. For example, when you push a lawn mower at constant speed on level ground, your work done is removed by the work of friction, and the mower has a constant energy.

### Applying Energy Conservation with Nonconservative Forces

When no change in potential energy occurs, applying  amounts to applying the work-energy theorem by setting the change in kinetic energy to be equal to the net work done on the system, which in the most general case includes both conservative and nonconservative forces. But when seeking instead to find a change in total mechanical energy in situations that involve changes in both potential and kinetic energy, the previous equation

says that you can start by finding the change in mechanical energy that would have resulted from just the conservative forces, including the potential energy changes, and add to it the work done, with the proper sign, by any nonconservative forces involved.

### Test Prep for AP Courses

### Section Summary

1. A nonconservative force is one for which work depends on the path.
2. Friction is an example of a nonconservative force that changes mechanical energy into thermal energy.
3. Work  done by a nonconservative force changes the mechanical energy of a system. In equation form,   or, equivalently, .
4. When both conservative and nonconservative forces act, energy conservation can be applied and used to calculate motion in terms of the known potential energies of the conservative forces and the work done by nonconservative forces, instead of finding the net work from the net force, or having to directly apply Newton’s laws.

### Problems & Exercises


# Work, Energy, and Energy Resources

## Conservation of Energy

### Learning Objectives

By the end of this section, you will be able to:
1. Explain the law of the conservation of energy.
2. Describe some of the many forms of energy.
3. Define efficiency of an energy conversion process as the fraction left as useful energy or work, rather than being transformed, for example, into thermal energy.

### Law of Conservation of Energy

Energy, as we have noted, is conserved, making it one of the most important physical quantities in nature. The law of conservation of energy can be stated as follows:
Total energy is constant in any process. It may change in form or be transferred from one system to another, but the total remains the same.
We have explored some forms of energy and some ways it can be transferred from one system to another. This exploration led to the definition of two major types of energy—mechanical energy  and energy transferred via work done by nonconservative forces . But energy takes many other forms, manifesting itself in many different ways, and we need to be able to deal with all of these before we can write an equation for the above general statement of the conservation of energy.

### Other Forms of Energy than Mechanical Energy

At this point, we deal with all other forms of energy by lumping them into a single group called  other energy (). Then we can state the conservation of energy in equation form as
All types of energy and work can be included in this very general statement of conservation of energy. Kinetic energy is , work done by a conservative force is represented by , work done by nonconservative forces is , and all other energies are included as . This equation applies to all previous examples; in those situations  was constant, and so it subtracted out and was not directly considered.
When does  play a role? One example occurs when a person eats. Food is oxidized with the release of carbon dioxide, water, and energy. Some of this chemical energy is converted to kinetic energy when the person moves, to potential energy when the person changes altitude, and to thermal energy (another form of ).

### Some of the Many Forms of Energy

What are some other forms of energy? You can probably name a number of forms of energy not yet discussed. Many of these will be covered in later chapters, but let us detail a few here. Electrical energy is a common form that is converted to many other forms and does work in a wide range of practical situations. Fuels, such as gasoline and food, carry chemical energy that can be transferred to a system through oxidation. Chemical fuel can also produce electrical energy, such as in batteries. Batteries can in turn produce light, which is a very pure form of energy. Most energy sources on Earth are in fact stored energy from the energy we receive from the Sun. We sometimes refer to this as radiant energy, or electromagnetic radiation, which includes visible light, infrared, and ultraviolet radiation. Nuclear energy comes from processes that convert measurable amounts of mass into energy. Nuclear energy is transformed into the energy of sunlight, into electrical energy in power plants, and into the energy of the heat transfer and blast in weapons. Atoms and molecules inside all objects are in random motion. This internal mechanical energy from the random motions is called thermal energy, because it is related to the temperature of the object. These and all other forms of energy can be converted into one another and can do work.
 gives the amount of energy stored, used, or released from various objects and in various phenomena. The range of energies and the variety of types and situations is impressive.

### Transformation of Energy

The transformation of energy from one form into others is happening all the time. The chemical energy in food is converted into thermal energy through metabolism; light energy is converted into chemical energy through photosynthesis. In a larger example, the chemical energy contained in coal is converted into thermal energy as it burns to turn water into steam in a boiler. This thermal energy in the steam in turn is converted to mechanical energy as it spins a turbine, which is connected to a generator to produce electrical energy. (In all of these examples, not all of the initial energy is converted into the forms mentioned. This important point is discussed later in this section.)
Another example of energy conversion occurs in a solar cell. Sunlight impinging on a solar cell (see ) produces electricity, which in turn can be used to run an electric motor. Energy is converted from the primary source of solar energy into electrical energy and then into mechanical energy.

### Efficiency

Even though energy is conserved in an energy conversion process, the output of useful energy or work will be less than the energy input. The efficiency  of an energy conversion process is defined as
 lists some efficiencies of mechanical devices and human activities. In a coal-fired power plant, for example, about 40% of the chemical energy in the coal becomes useful electrical energy. The other 60% transforms into other (perhaps less useful) energy forms, such as thermal energy, which is then released to the environment through combustion gases and cooling towers.

### Test Prep for AP Courses

### Section Summary

1. The law of conservation of energy states that the total energy is constant in any process. Energy may change in form or be transferred from one system to another, but the total remains the same.
2. When all forms of energy are considered, conservation of energy is written in equation form as , where  is all other forms of energy besides mechanical energy.
3. Commonly encountered forms of energy include electric energy, chemical energy, radiant energy, nuclear energy, and thermal energy.
4. Energy is often utilized to do work, but it is not possible to convert all the energy of a system to work.
5. The efficiency  of a machine or human is defined to be , where  is useful work output and  is the energy consumed.

### Conceptual Questions

### Problems & Exercises


# Work, Energy, and Energy Resources

## Power

### Learning Objectives

By the end of this section, you will be able to:
1. Calculate power by calculating changes in energy over time.
2. Examine power consumption and calculations of the cost of energy consumed.

### What is Power?

Power—the word conjures up many images: a professional football player muscling aside his opponent, a dragster roaring away from the starting line, a volcano blowing its lava into the atmosphere, or a rocket blasting off, as in .
These images of power have in common the rapid performance of work, consistent with the scientific definition of power () as the rate at which work is done.
Because work is energy transfer, power is also the rate at which energy is expended. A 60-W light bulb, for example, expends 60 J of energy per second. Great power means a large amount of work or energy developed in a short time. For example, when a powerful car accelerates rapidly, it does a large amount of work and consumes a large amount of fuel in a short time.

### Calculating Power from Energy

It is impressive that this woman’s useful power output is slightly less than 1 horsepower ! People can generate more than a horsepower with their leg muscles for short periods of time by rapidly converting available blood sugar and oxygen into work output. (A horse can put out 1 hp for hours on end.) Once oxygen is depleted, power output decreases and the person begins to breathe rapidly to obtain oxygen to metabolize more food—this is known as the aerobic stage of exercise. If the woman climbed the stairs slowly, then her power output would be much less, although the amount of work done would be the same.

### Examples of Power

Examples of power are limited only by the imagination, because there are as many types as there are forms of work and energy. (See  for some examples.) Sunlight reaching Earth’s surface carries a maximum power of about 1.3 kilowatts per square meter  A tiny fraction of this is retained by Earth over the long term. Our consumption rate of fossil fuels is far greater than the rate at which they are stored, so it is inevitable that they will be depleted. Power implies that energy is transferred, perhaps changing form. It is never possible to change one form completely into another without losing some of it as thermal energy. For example, a 60-W incandescent bulb converts only 5 W of electrical power to light, with 55 W dissipating into thermal energy. Furthermore, the typical electric power plant converts only 35 to 40% of its fuel into electricity. The remainder becomes a huge amount of thermal energy that must be dispersed as heat transfer, as rapidly as it is created. A coal-fired power plant may produce 1000 megawatts; 1 megawatt (MW) is  of electric power. But the power plant consumes chemical energy at a rate of about 2500 MW, creating heat transfer to the surroundings at a rate of 1500 MW. (See .)

### Power and Energy Consumption

We usually have to pay for the energy we use. It is interesting and easy to estimate the cost of energy for an electrical appliance if its power consumption rate and time used are known. The higher the power consumption rate and the longer the appliance is used, the greater the cost of that appliance. The power consumption rate is , where  is the energy supplied by the electricity company. So the energy consumed over a time  is
Electricity bills state the energy used in units of kilowatt-hours  which is the product of power in kilowatts and time in hours. This unit is convenient because electrical power consumption at the kilowatt level for hours at a time is typical.
The motivation to save energy has become more compelling with its ever-increasing price. Armed with the knowledge that energy consumed is the product of power and time, you can estimate costs for yourself and make the necessary value judgments about where to save energy. Either power or time must be reduced. It is most cost-effective to limit the use of high-power devices that normally operate for long periods of time, such as water heaters and air conditioners. This would not include relatively high power devices like toasters, because they are on only a few minutes per day. It would also not include electric clocks, in spite of their 24-hour-per-day usage, because they are very low power devices. It is sometimes possible to use devices that have greater efficiencies—that is, devices that consume less power to accomplish the same task. One example is the compact fluorescent light bulb, which produces over four times more light per watt of power consumed than its incandescent cousin.
Modern civilization depends on energy, but current levels of energy consumption and production are not sustainable. The likelihood of a link between global warming and fossil fuel use (with its concomitant production of carbon dioxide), has made reduction in energy use as well as a shift to non-fossil fuels of the utmost importance. Even though energy in an isolated system is a conserved quantity, the final result of most energy transformations is waste heat transfer to the environment, which is no longer useful for doing work. As we will discuss in more detail in Thermodynamics, the potential for energy to produce useful work has been “degraded” in the energy transformation.

### Section Summary

1. Power is the rate at which work is done, or in equation form, for the average power  for work  done over a time , .
2. The SI unit for power is the watt  (W), where .
3. The power of many devices such as electric motors is also often expressed in horsepower (hp), where .

### Conceptual Questions

### Problems & Exercises


# Work, Energy, and Energy Resources

## Work, Energy, and Power in Humans

### Learning Objectives

By the end of this section, you will be able to:
1. Explain the human body’s consumption of energy when at rest vs. when engaged in activities that do useful work.
2. Calculate the conversion of chemical energy in food into useful work.

### Energy Conversion in Humans

Our own bodies, like all living organisms, are energy conversion machines. Conservation of energy implies that the chemical energy stored in food is converted into work, thermal energy, and/or stored as chemical energy in fatty tissue. (See .) The fraction going into each form depends both on how much we eat and on our level of physical activity. If we eat more than is needed to do work and stay warm, the remainder goes into body fat.

### Power Consumed at Rest

The rate at which the body uses food energy to sustain life and to do different activities is called the metabolic rate. The total energy conversion rate of a person at rest is called the basal metabolic rate (BMR) and is divided among various systems in the body, as shown in . The largest fraction goes to the liver and spleen, with the brain coming next. Of course, during vigorous exercise, the energy consumption of the skeletal muscles and heart increase markedly. About 75% of the calories burned in a day go into these basic functions. The BMR is a function of age, gender, total body weight, and amount of muscle mass (which burns more calories than body fat). Athletes have a greater BMR due to this last factor.
Energy consumption is directly proportional to oxygen consumption because the digestive process is basically one of oxidizing food. We can measure the energy people use during various activities by measuring their oxygen use. (See .) Approximately 20 kJ of energy are produced for each liter of oxygen consumed, independent of the type of food.  shows energy and oxygen consumption rates (power expended) for a variety of activities.

### Power of Doing Useful Work

Work done by a person is sometimes called useful work, which is work done on the outside world, such as lifting weights. Useful work requires a force exerted through a distance on the outside world, and so it excludes internal work, such as that done by the heart when pumping blood. Useful work does include that done in climbing stairs or accelerating to a full run, because these are accomplished by exerting forces on the outside world. Forces exerted by the body are nonconservative, so that they can change the mechanical energy () of the system worked upon, and this is often the goal. A baseball player throwing a ball, for example, increases both the ball’s kinetic and potential energy.
If a person needs more energy than they consume, such as when doing vigorous work, the body must draw upon the chemical energy stored in fat. So exercise can be helpful in losing fat. However, the amount of exercise needed to produce a loss in fat, or to burn off extra calories consumed that day, can be large, as  illustrates.
All bodily functions, from thinking to lifting weights, require energy. (See .) The many small muscle actions accompanying all quiet activity, from sleeping to head scratching, ultimately become thermal energy, as do less visible muscle actions by the heart, lungs, and digestive tract. Shivering, in fact, is an involuntary response to low body temperature that pits muscles against one another to produce thermal energy in the body (and do no work). The kidneys and liver consume a surprising amount of energy, but the biggest surprise of all is that a full 25% of all energy consumed by the body is used to maintain electrical potentials in all living cells. (Nerve cells use this electrical potential in nerve impulses.) This bioelectrical energy ultimately becomes mostly thermal energy, but some is utilized to power chemical processes such as in the kidneys and liver, and in fat production.

### Section Summary

1. The human body converts energy stored in food into work, thermal energy, and/or chemical energy that is stored in fatty tissue.
2. The rate at which the body uses food energy to sustain life and to do different activities is called the metabolic rate, and the corresponding rate when at rest is called the basal metabolic rate (BMR)
3. The energy included in the basal metabolic rate is divided among various systems in the body, with the largest fraction going to the liver and spleen, and the brain coming next.
4. About 75% of food calories are used to sustain basic body functions included in the basal metabolic rate.
5. The energy consumption of people during various activities can be determined by measuring their oxygen use, because the digestive process is basically one of oxidizing food.

### Conceptual Questions

### Problems & Exercises


# Work, Energy, and Energy Resources

## World Energy Use

### Learning Objectives

By the end of this section, you will be able to:
1. Describe the distinction between renewable and nonrenewable energy sources.
2. Explain why the inevitable conversion of energy to less useful forms makes it necessary to conserve energy resources.
Energy is an important ingredient in all phases of society. We live in a very interdependent world, and access to adequate and reliable energy resources is crucial for economic growth and for maintaining the quality of our lives. But current levels of energy consumption and production are not sustainable. Depending on the data source, estimates indicate that about 31–35% of the world’s energy comes from oil, and much of that goes to transportation uses. This is a reduction by a few percentage points from ten years ago. Oil prices are dependent as much upon new (or foreseen) discoveries as they are upon political events and situations around the world. The U.S., with 4.25% of the world’s population, consumes 21% of the world’s oil production per year.

### Renewable and Nonrenewable Energy Sources

The principal energy resources used in the world are shown in . The fuel mix has changed over the years but now is dominated by oil, although natural gas and solar contributions are increasing. Renewable forms of energy are those sources that cannot be used up, such as water, wind, solar, and biomass. About 85% of our energy comes from nonrenewable fossil fuels—oil, natural gas, coal. The likelihood of a link between global warming and fossil fuel use, with its production of carbon dioxide through combustion, has made, in the eyes of many scientists, a shift to non-fossil fuels of utmost importance—but it will not be easy.

### The World’s Growing Energy Needs

World energy consumption continues to rise, especially in the developing countries. (See .) Global demand for energy has tripled in the past 50 years and might triple again in the next 30 years. While much of this growth will come from the rapidly booming economies of China and India, many of the developed countries, especially those in Europe, are hoping to meet their energy needs by expanding the use of renewable sources. Although presently only a small percentage, renewable energy is growing very fast, especially wind energy. For example, Germany plans to meet 65% of its power and 30% of its overall energy needs with renewable resources by the year 2030. (See .) Energy is a key constraint in the rapid economic growth of China and India. In 2003, China surpassed Japan as the world’s second largest consumer of oil. However, over 1/3 of this is imported. Unlike most Western countries, coal dominates the commercial energy resources of China, accounting for 2/3 of its energy consumption. In 2009 China surpassed the United States as the largest generator of . In India, the main energy resources are biomass (wood and dung) and coal. Half of India’s oil is imported. About 70% of India’s electricity is generated by highly polluting coal. Yet there are sizeable strides being made in renewable energy. India has a rapidly growing wind energy base, and it has the largest solar cooking program in the world. China has invested substantially in building solar collection farms as well as hydroelectric plants.
 displays the 2020 commercial energy mix by country for some of the prime energy users in the world. While non-renewable sources dominate, some countries get a sizeable percentage of their electricity from renewable resources. For example, about two-thirds of New Zealand’s electricity demand is met by hydroelectric. Only 10% of the U.S. electricity is generated by renewable resources, primarily hydroelectric. It is difficult to determine total sources and consumers of energy in many countries, and estimates vary somewhat by data source and type of measurement.

### Energy and Economic Well-being

Economic well-being is dependent upon energy use, and in most countries higher standards of living, as measured by GDP (gross domestic product) per capita, are matched by higher levels of energy consumption per capita. This is borne out in . Increased efficiency of energy use will change this dependency. A global problem is balancing energy resource development against the harmful effects upon the environment in its extraction and use.
New and diversified energy sources do, however, greatly increase economic opportunity and stability. First, the extensive employment opportunities in renewable energy make it one of the most sustainable and secure fields to enter. Second, renewable energy provides countries and localities with increased levels of resiliency in the face of natural disasters, conflict, or other disruptions. The 21st century has already seen major economic impacts from energy disruptions: Hurricane Katrina, Superstorm Sandy, various wildfires, Hurricane Maria, and the 2021 Texas Winter Storm demonstrate the vulnerability of United States power systems. Diversifying energy sources through renewables and other fossil-fuel alternatives brings power grids and transportation systems back online much more quickly, saving lives and enabling a more swift return to economic operations. And as critical emerging information infrastructure, such as data centers, requires more of the world's energy, supplying those growing systems during normal operations and crises will be increasingly important.

### Conserving Energy

As we finish this chapter on energy and work, it is relevant to draw some distinctions between two sometimes misunderstood terms in the area of energy use. As has been mentioned elsewhere, the “law of the conservation of energy” is a very useful principle in analyzing physical processes. It is a statement that cannot be proven from basic principles, but is a very good bookkeeping device, and no exceptions have ever been found. It states that the total amount of energy in an isolated system will always remain constant. Related to this principle, but remarkably different from it, is the important philosophy of energy conservation. This concept has to do with seeking to decrease the amount of energy used by an individual or group through (1) reduced activities (e.g., turning down thermostats, driving fewer kilometers) and/or (2) increasing conversion efficiencies in the performance of a particular task—such as developing and using more efficient room heaters, cars that have greater miles-per-gallon ratings, energy-efficient compact fluorescent lights, etc.
Since energy in an isolated system is not destroyed or created or generated, one might wonder why we need to be concerned about our energy resources, since energy is a conserved quantity. The problem is that the final result of most energy transformations is waste heat transfer to the environment and conversion to energy forms no longer useful for doing work. To state it in another way, the potential for energy to produce useful work has been “degraded” in the energy transformation. (This will be discussed in more detail in Thermodynamics.)

### Section Summary

1. The relative use of different fuels to provide energy has changed over the years, but fuel use is currently dominated by oil, although natural gas and solar contributions are increasing.

2. Although non-renewable sources dominate, some countries meet a sizeable percentage of their electricity needs from renewable resources.
3. The United States obtains only about 10% of its energy from renewable sources, mostly hydroelectric power.
4. Economic well-being is dependent upon energy use, and in most countries higher standards of living, as measured by GDP (Gross Domestic Product) per capita, are matched by higher levels of energy consumption per capita.
5. Even though, in accordance with the law of conservation of energy, energy can never be created or destroyed, energy that can be used to do work is always partly converted to less useful forms, such as waste heat to the environment, in all of our uses of energy for practical purposes.

### Conceptual Questions

### Problems & Exercises


# Linear Momentum and Collisions

## Connection for AP® courses

In this chapter, you will learn about the concept of momentum and the relationship between momentum and force (both vector quantities) applied over a time interval. Have you ever considered why a glass dropped on a tile floor will often break, but a glass dropped on carpet will often remain intact? Both involve changes in momentum, but the actual collision with the floor is different in each case, just as an automobile collision without the benefit of an airbag can have a significantly different outcome than one with an airbag.
You will learn that the interaction of objects (like a glass and the floor or two automobiles) results in forces, which in turn result in changes in the momentum of each object. At the same time, you will see how the law of momentum conservation can be applied to a system to help determine the outcome of a collision.
The content in this chapter supports:
Big Idea 3 The interactions of an object with other objects can be described by forces.
Enduring Understanding 3.D A force exerted on an object can change the momentum of the object.
Essential Knowledge 3.D.2 The change in momentum of an object occurs over a time interval.
Big Idea 4: Interactions between systems can result in changes in those systems.
Enduring Understanding 4.B Interactions with other objects or systems can change the total linear momentum of a system.
Essential Knowledge 4.B.1 The change in linear momentum for a constant-mass system is the product of the mass of the system and the change in velocity of the center of mass.
Big Idea 5 Changes that occur as a result of interactions are constrained by conservation laws.
Enduring Understanding 5.A Certain quantities are conserved, in the sense that the changes of those quantities in a given system are always equal to the transfer of that quantity to or from the system by all possible interactions with other systems.
Essential Knowledge 5.A.2 For all systems under all circumstances, energy, charge, linear momentum, and angular momentum are conserved.
Essential Knowledge 5.D.1 In a collision between objects, linear momentum is conserved. In an elastic collision, kinetic energy is the same before and after.
Essential Knowledge 5.D.2 In a collision between objects, linear momentum is conserved. In an inelastic collision, kinetic energy is not the same before and after the collision.


# Linear Momentum and Collisions

## Linear Momentum and Force

### Learning Objectives

By the end of this section, you will be able to:
1. Define linear momentum.
2. Explain the relationship between momentum and force.
3. State Newton’s second law of motion in terms of momentum.
4. Calculate momentum given mass and velocity.

### Linear Momentum

The scientific definition of linear momentum is consistent with most people’s intuitive understanding of momentum: a large, fast-moving object has greater momentum than a smaller, slower object. Linear momentum is defined as the product of a system’s mass multiplied by its velocity. In symbols, linear momentum is expressed as
Momentum is directly proportional to the object’s mass and also its velocity. Thus the greater an object’s mass or the greater its velocity, the greater its momentum. Momentum  is a vector having the same direction as the velocity . The SI unit for momentum is .

### Momentum and Newton’s Second Law

The importance of momentum, unlike the importance of energy, was recognized early in the development of classical physics. Momentum was deemed so important that it was called the “quantity of motion.” Newton actually stated his second law of motion in terms of momentum: The net external force equals the change in momentum of a system divided by the time over which it changes. Using symbols, this law is
where  is the net external force,  is the change in momentum, and  is the change in time.
This statement of Newton’s second law of motion includes the more familiar  as a special case. We can derive this form as follows. First, note that the change in momentum  is given by
If the mass of the system is constant, then
So that for constant mass, Newton’s second law of motion becomes
Because , we get the familiar equation
when the mass of the system is constant.
Newton’s second law of motion stated in terms of momentum is more generally applicable because it can be applied to systems where the mass is changing, such as rockets, as well as to systems of constant mass. We will consider systems with varying mass in some detail; however, the relationship between momentum and force remains useful when mass is constant, such as in the following example.

### Test Prep for AP Courses

### Section Summary

1. Linear momentum (momentum for brevity) is defined as the product of a system’s mass multiplied by its velocity.
2. In symbols, linear momentum  is defined to be

    where  is the mass of the system and  is its velocity.
3. The SI unit for momentum is .
4. Newton’s second law of motion in terms of momentum states that the net external force equals the change in momentum of a system divided by the time over which it changes.
5. In symbols, Newton’s second law of motion is defined to be
     is the net external force,  is the change in momentum, and  is the change time.

### Conceptual Questions

### Problems & Exercises


# Linear Momentum and Collisions

## Impulse

### Learning Objectives

By the end of this section, you will be able to:
1. Define impulse.
2. Describe effects of impulses in everyday life.
3. Determine the average effective force using graphical representation.
4. Calculate average force and impulse given mass, velocity, and time.
The effect of a force on an object depends on how long it acts, as well as how great the force is. In , a very large force acting for a short time had a great effect on the momentum of the tennis ball. A small force could cause the same change in momentum, but it would have to act for a much longer time. For example, if the ball were thrown upward, the gravitational force (which is much smaller than the tennis racquet’s force) would eventually reverse the momentum of the ball. Quantitatively, the effect we are talking about is the change in momentum .
By rearranging the equation  to be
we can see how the change in momentum equals the average net external force multiplied by the time this force acts. The quantity  is given the name impulse. Impulse is the same as the change in momentum.
Our definition of impulse includes an assumption that the force is constant over the time interval . Forces are usually not constant. Forces vary considerably even during the brief time intervals considered. It is, however, possible to find an average effective force  that produces the same result as the corresponding time-varying force.  shows a graph of what an actual force looks like as a function of time for a ball bouncing off the floor. The area under the curve has units of momentum and is equal to the impulse or change in momentum between times  and . That area is equal to the area inside the rectangle bounded by , , and . Thus the impulses and their effects are the same for both the actual and effective forces.

### Test Prep for AP Courses

### Section Summary

1. Impulse, or change in momentum, equals the average net external force multiplied by the time this force acts:

2. Forces are usually not constant over a period of time.

### Conceptual Questions

### Problems & Exercises


# Linear Momentum and Collisions

## Conservation of Momentum

### Learning Objectives

By the end of this section, you will be able to:
1. Describe the principle of conservation of momentum.
2. Derive an expression for the conservation of momentum.
3. Explain conservation of momentum with examples.
4. Explain the principle of conservation of momentum as it relates to atomic and subatomic particles.
Momentum is an important quantity because it is conserved. Yet it was not conserved in the examples in Impulse and Linear Momentum and Force, where large changes in momentum were produced by forces acting on the system of interest. Under what circumstances is momentum conserved?
The answer to this question entails considering a sufficiently large system. It is always possible to find a larger system in which total momentum is constant, even if momentum changes for components of the system. If a football player runs into the goalpost in the end zone, there will be a force on him that causes him to bounce backward. The backward momentum felt by an object or person exerting force on another object is often called a recoil. However, the Earth also recoils—conserving momentum—because of the force applied to it through the goalpost. Because Earth is many orders of magnitude more massive than the player, its recoil is immeasurably small and can be neglected in any practical sense, but it is real nevertheless.
Consider what happens if the masses of two colliding objects are more similar than the masses of a football player and Earth—for example, one car bumping into another, as shown in . Both cars are coasting in the same direction when the lead car (labeled  is bumped by the trailing car (labeled  The only unbalanced force on each car is the force of the collision. (Assume that the effects due to friction are negligible.) Car 1 slows down as a result of the collision, losing some momentum, while car 2 speeds up and gains some momentum. We shall now show that the total momentum of the two-car system remains constant.
Using the definition of impulse, the change in momentum of car 1 is given by
where  is the force on car 1 due to car 2, and  is the time the force acts (the duration of the collision). Intuitively, it seems obvious that the collision time is the same for both cars, but it is only true for objects traveling at ordinary speeds. This assumption must be modified for objects travelling near the speed of light, without affecting the result that momentum is conserved.
Similarly, the change in momentum of car 2 is
where  is the force on car 2 due to car 1, and we assume the duration of the collision  is the same for both cars. We know from Newton’s third law that , and so
Thus, the changes in momentum are equal and opposite, and
Because the changes in momentum add to zero, the total momentum of the two-car system is constant. That is,
where  and  are the momenta of cars 1 and 2 after the collision. (We often use primes to denote the final state.)
This result—that momentum is conserved—has validity far beyond the preceding one-dimensional case. It can be similarly shown that total momentum is conserved for any isolated system, with any number of objects in it. In equation form, the conservation of momentum principle for an isolated system is written
or
where  is the total momentum (the sum of the momenta of the individual objects in the system) and  is the total momentum some time later. (Recall in Uniform Circular Motion and Gravitation you learned that the center of mass of a system of objects is the effective average location of the mass of the system.  The total momentum can be shown to be the momentum of the center of mass of the system.) An isolated system is defined to be one for which the net external force is zero
Perhaps an easier way to see that momentum is conserved for an isolated system is to consider Newton’s second law in terms of momentum, . For an isolated system,

; thus, , and
 is constant.
We have noted that the three length dimensions in nature—, , and —are independent, and it is interesting to note that momentum can be conserved in different ways along each dimension. For example, during projectile motion and where air resistance is negligible, momentum is conserved in the horizontal direction because horizontal forces are zero and momentum is unchanged. But along the vertical direction, the net vertical force is not zero and the momentum of the projectile is not conserved. (See .) However, if the momentum of the projectile-Earth system is considered in the vertical direction, we find that the total momentum is conserved.
The conservation of momentum principle can be applied to systems as different as a comet striking Earth and a gas containing huge numbers of atoms and molecules. Conservation of momentum is violated only when the net external force is not zero. But another larger system can always be considered in which momentum is conserved by simply including the source of the external force. For example, in the collision of two cars considered above, the two-car system conserves momentum while each one-car system does not.

### Subatomic Collisions and Momentum

The conservation of momentum principle not only applies to the macroscopic objects, it is also essential to our explorations of atomic and subatomic particles. Giant machines hurl subatomic particles at one another, and researchers evaluate the results by assuming conservation of momentum (among other things).
On the small scale, we find that particles and their properties are invisible to the naked eye but can be measured with our instruments, and models of these subatomic particles can be constructed to describe the results. Momentum is found to be a property of all subatomic particles including massless particles such as photons that compose light. Momentum being a property of particles hints that momentum may have an identity beyond the description of an object’s mass multiplied by the object’s velocity. Indeed, momentum relates to wave properties and plays a fundamental role in what measurements are taken and how we take these measurements. Furthermore, we find that the conservation of momentum principle is valid when considering systems of particles. We use this principle to analyze the masses and other properties of previously undetected particles, such as the nucleus of an atom and the existence of quarks that make up particles of nuclei.  below illustrates how a particle scattering backward from another implies that its target is massive and dense. Experiments seeking evidence that quarks make up protons (one type of particle that makes up nuclei) scattered high-energy electrons off of protons (nuclei of hydrogen atoms). Electrons occasionally scattered straight backward in a manner that implied a very small and very dense particle makes up the proton—this observation is considered nearly direct evidence of quarks. The analysis was based partly on the same conservation of momentum principle that works so well on the large scale.

### Test Prep for AP Courses

### Section Summary

1. The conservation of momentum principle is written
 or is the initial total momentum and  is the total momentum some time later.
2. An isolated system is defined to be one for which the net external force is zero
3. During projectile motion and where air resistance is negligible, momentum is conserved in the horizontal direction because horizontal forces are zero.
4. Conservation of momentum applies only when the net external force is zero.
5. The conservation of momentum principle is valid when considering systems of particles.

### Conceptual Questions

### Problems & Exercises


# Linear Momentum and Collisions

## Elastic Collisions in One Dimension

### Learning Objectives

By the end of this section, you will be able to:
1. Describe an elastic collision of two objects in one dimension.
2. Define internal kinetic energy.
3. Derive an expression for conservation of internal kinetic energy in a one dimensional collision.
4. Determine the final velocities in an elastic collision given masses and initial velocities.
Let us consider various types of two-object collisions. These collisions are the easiest to analyze, and they illustrate many of the physical principles involved in collisions. The conservation of momentum principle is very useful here, and it can be used whenever the net external force on a system is zero.
We start with the elastic collision of two objects moving along the same line—a one-dimensional problem. An elastic collision is one that also conserves internal kinetic energy. Internal kinetic energy is the sum of the kinetic energies of the objects in the system.  illustrates an elastic collision in which internal kinetic energy and momentum are conserved.
Truly elastic collisions can only be achieved with subatomic particles, such as electrons striking nuclei. Macroscopic collisions can be very nearly, but not quite, elastic—some kinetic energy is always converted into other forms of energy such as heat transfer due to friction and sound. One macroscopic collision that is nearly elastic is that of two steel blocks on ice. Another nearly elastic collision is that between two carts with spring bumpers on an air track. Icy surfaces and air tracks are nearly frictionless, more readily allowing nearly elastic collisions on them.

Now, to solve problems involving one-dimensional elastic collisions between two objects we can use the equations for conservation of momentum and conservation of internal kinetic energy. First, the equation for conservation of momentum for two objects in a one-dimensional collision is
or
where the primes (') indicate values after the collision. By definition, an elastic collision conserves internal kinetic energy, and so the sum of kinetic energies before the collision equals the sum after the collision. Thus,
expresses the equation for conservation of internal kinetic energy in a one-dimensional collision.

### Test Prep for AP Courses

### Section Summary

1. An elastic collision is one that conserves internal kinetic energy.
2. Conservation of kinetic energy and momentum together allow the final velocities to be calculated in terms of initial velocities and masses in one dimensional two-body collisions.

### Conceptual Questions

### Problems & Exercises


# Linear Momentum and Collisions

## Inelastic Collisions in One Dimension

### Learning Objectives

By the end of this section, you will be able to:
1. Define inelastic collision.
2. Explain perfectly inelastic collision.
3. Apply an understanding of collisions to sports.
4. Determine recoil velocity and loss in kinetic energy given mass and initial velocity.
We have seen that in an elastic collision, internal kinetic energy is conserved. An inelastic collision is one in which the internal kinetic energy changes (it is not conserved). This lack of conservation means that the forces between colliding objects may remove or add internal kinetic energy. Work done by internal forces may change the forms of energy within a system. For inelastic collisions, such as when colliding objects stick together, this internal work may transform some internal kinetic energy into heat transfer. Or it may convert stored energy into internal kinetic energy, such as when exploding bolts separate a satellite from its launch vehicle.
 shows an example of an inelastic collision. Two objects that have equal masses head toward one another at equal speeds and then stick together. Their total internal kinetic energy is initially
.  The two objects come to rest after sticking together, conserving momentum. But the internal kinetic energy is zero after the collision. A collision in which the objects stick together is sometimes called a perfectly inelastic collision because it reduces internal kinetic energy more than does any other type of inelastic collision. In fact, such a collision reduces internal kinetic energy to the minimum it can have while still conserving momentum.

### Test Prep for AP Courses

### Section Summary

1. An inelastic collision is one in which the internal kinetic energy changes (it is not conserved).
2. A collision in which the objects stick together is sometimes called perfectly inelastic because it reduces internal kinetic energy more than does any other type of inelastic collision.
3. Sports science and technologies also use physics concepts such as momentum and rotational motion and vibrations.

### Conceptual Questions

### Problems & Exercises


# Linear Momentum and Collisions

## Collisions of Point Masses in Two Dimensions

### Learning Objectives

By the end of this section, you will be able to:
1. Discuss two dimensional collisions as an extension of one dimensional analysis.
2. Define point masses.
3. Derive an expression for conservation of momentum along x-axis and y-axis.
4. Describe elastic collisions of two objects with equal mass.
5. Determine the magnitude and direction of the final velocity given initial velocity, and scattering angle.
In the previous two sections, we considered only one-dimensional collisions; during such collisions, the incoming and outgoing velocities are all along the same line. But what about collisions, such as those between billiard balls, in which objects scatter to the side? These are two-dimensional collisions, and we shall see that their study is an extension of the one-dimensional analysis already presented. The approach taken (similar to the approach in discussing two-dimensional kinematics and dynamics) is to choose a convenient coordinate system and resolve the motion into components along perpendicular axes. Resolving the motion yields a pair of one-dimensional problems to be solved simultaneously.
One complication arising in two-dimensional collisions is that the objects might rotate before or after their collision. For example, if two ice skaters hook arms as they pass by one another, they will spin in circles. We will not consider such rotation until later, and so for now we arrange things so that no rotation is possible. To avoid rotation, we consider only the scattering of point masses—that is, structureless particles that cannot rotate or spin.
We start by assuming that

, so that momentum

 is conserved. The simplest collision is one in which one of the particles is initially at rest. (See .) The best choice for a coordinate system is one with an axis parallel to the velocity of the incoming particle, as shown in . Because momentum is conserved, the components of momentum along the

- and -axes

will also be conserved, but with the chosen coordinate system,

 is initially zero and

is the momentum of the incoming particle. Both facts simplify the analysis. (Even with the simplifying assumptions of point masses, one particle initially at rest, and a convenient coordinate system, we still gain new insights into nature from the analysis of two-dimensional collisions.)
Along the -axis, the equation for conservation of momentum is
Where the subscripts denote the particles and axes and the primes denote the situation after the collision. In terms of masses and velocities, this equation is
But because particle 2 is initially at rest, this equation becomes
The components of the velocities along the -axis have the form . Because particle 1 initially moves along the -axis, we find .
Conservation of momentum along the -axis gives the following equation:
where  and  are as shown in .
Along the -axis, the equation for conservation of momentum is
or
But  is zero, because particle 1 initially moves along the -axis. Because particle 2 is initially at rest,  is also zero. The equation for conservation of momentum along the -axis becomes
The components of the velocities along the -axis have the form .
Thus, conservation of momentum along the -axis gives the following equation:
The equations of conservation of momentum along the -axis and -axis are very useful in analyzing two-dimensional collisions of particles, where one is originally stationary (a common laboratory situation). But two equations can only be used to find two unknowns, and so other data may be necessary when collision experiments are used to explore nature at the subatomic level.

### Elastic Collisions of Two Objects with Equal Mass

Some interesting situations arise when the two colliding objects have equal mass and the collision is elastic. This situation is nearly the case with colliding billiard balls, and precisely the case with some subatomic particle collisions. We can thus get a mental image of a collision of subatomic particles by thinking about billiards (or pool). (Refer to  for masses and angles.) First, an elastic collision conserves internal kinetic energy. Again, let us assume object 2  is initially at rest. Then, the internal kinetic energy before and after the collision of two objects that have equal masses is
Because the masses are equal, . Algebraic manipulation (left to the reader) of conservation of momentum in the - and -directions can show that
(Remember that  is negative here.) The two preceding equations can both be true only if
There are three ways that this term can be zero. They are
1. : head-on collision; incoming ball stops
2. : no collision; incoming ball continues unaffected
3. : angle of separation  is  after the collision
All three of these ways are familiar occurrences in billiards and pool, although most of us try to avoid the second. If you play enough pool, you will notice that the angle between the balls is very close to  after the collision, although it will vary from this value if a great deal of spin is placed on the ball. (Large spin carries in extra energy and a quantity called angular momentum, which must also be conserved.) The assumption that the scattering of billiard balls is elastic is reasonable based on the correctness of the three results it produces. This assumption also implies that, to a good approximation, momentum is conserved for the two-ball system in billiards and pool. The problems below explore these and other characteristics of two-dimensional collisions.

### Test Prep for AP Courses

### Section Summary

1. The approach to two-dimensional collisions is to choose a convenient coordinate system and break the motion into components along perpendicular axes. Choose a coordinate system with the -axis parallel to the velocity of the incoming particle.
2.
Two-dimensional collisions of point masses where mass 2 is initially at rest conserve momentum along the initial direction of mass 1 (the -axis), stated by
 and along the direction perpendicular to the initial direction (the
-axis) stated by
.

3. The internal kinetic before and after the collision of two objects that have equal masses is
4. Point masses are structureless particles that cannot spin.

### Conceptual Questions

### Problems & Exercises