The concept of energy is one of the most important topics in science and engineering. In everyday life, we think of energy in terms of fuel for transportation and heating, electricity for lights and appliances, and foods for consumption. However, these ideas do not really define energy. They merely tell us that fuels are needed to do a job and that those fuels provide us with something we call energy.
The definitions of quantities such as position, velocity, acceleration, and force and
associated principles such as Newton’s second law have allowed us to solve a variety of problems. Some problems that could theoretically be solved with Newton’s laws, however, are very difficult in practice. These problems can be made much simpler with a different approach. In this and the following chapters, we will investigate this new approach, which will include definitions of quantities that may not be familiar to you. Other quantities may sound familiar, but they may have more specific meanings in physics than in everyday life. We begin this discussion by exploring the notion of energy.
Energy is present in the Universe in various forms. Every physical process that occurs
in the Universe involves energy and energy transfers or transformations. Unfortunately, despite its extreme importance, energy cannot be easily defined. The variables in previous chapters were relatively concrete; we have everyday experience with velocities and forces, for example. The notion of energy is more abstract, although we do have experiences with energy, such as running out of gasoline, or losing our electrical service if we forget to pay the utility bill.
The concept of energy can be applied to the dynamics of a mechanical system without resorting to Newton’s laws. This “energy approach” to describing motion is especially useful when the force acting on a particle is not constant; in such a case, the acceleration is not constant, and we cannot apply the constant acceleration equations that were developed in Chapter 2. Particles in nature are often subject to forces that vary with the particles’ positions. These forces include gravitational forces and the force exerted on an object attached to a spring. We shall describe techniques for treating such situations with the help of an important concept called conservation of energy. This approach extends well beyond physics, and can be applied to biological organisms, technological systems, and engineering situations.
Our problem-solving techniques presented in earlier chapters were based on the
motion of a particle or an object that could be modeled as a particle. This was called
the particle model. We begin our new approach by focusing our attention on a system and
developing techniques to be used in a system model.
7.1 Systems and Environments
In the system model mentioned above, we focus our attention on a small portion of the
Universe—the system—and ignore details of the rest of the Universe outside of the
system. A critical skill in applying the system model to problems is identifying the system.
A valid system may
• be a single object or particle
• be a collection of objects or particles
• be a region of space (such as the interior of an automobile engine combustion
• vary in size and shape (such as a rubber ball, which deforms upon striking a
Identifying the need for a system approach to solving a problem (as opposed to a
particle approach) is part of the “categorize” step in the General Problem-Solving
Strategy outlined in Chapter 2. Identifying the particular system and its nature is part
of the “analyze” step.
No matter what the particular system is in a given problem, there is a system
boundary, an imaginary surface (not necessarily coinciding with a physical surface)
that divides the Universe into the system and the environment surrounding the
As an example, imagine a force applied to an object in empty space. We can define
the object as the system. The force applied to it is an influence on the system from the
environment that acts across the system boundary. We will see how to analyze this situation
from a system approach in a subsequent section of this chapter.
Another example is seen in Example 5.10 (page 130). Here the system can be defined
as the combination of the ball, the cube, and the string. The influence from the
environment includes the gravitational forces on the ball and the cube, the normal
and friction forces on the cube, and the force exerted by the pulley on the string. The
forces exerted by the string on the ball and the cube are internal to the system and,
therefore, are not included as an influence from the environment.
We shall find that there are a number of mechanisms by which a system can be influenced by its environment. The first of these that we shall investigate is work.
7.2 Work Done by a Constant Force
Almost all the terms we have used thus far—velocity, acceleration, force, and so on—
convey a similar meaning in physics as they do in everyday life. Now, however, we encounter
a term whose meaning in physics is distinctly different from its everyday meaning—
Let us examine the situation in Figure 7.2, where an object undergoes a displacement
along a straight line while acted on by a constant force F that makes an angle Ɵ
with the direction of the displacement.
The work W done on a system by an agent exerting a constant force on the system is
the product of the magnitude F of the force, the magnitude r of the displacement
of the point of application of the force, and cos Ɵ, where Ɵ is the angle between the
force and displacement vectors:
The sign of the work also depends on the direction of F relative to r. The work
done by the applied force is positive when the projection of F ont vr is in the same
direction as the displacement. For example, when an object is lifted, the work done
by the applied force is positive because the direction of that force is upward, in the
same direction as the displacement of its point of application. When the projection
of F onto r is in the direction opposite the displacement, W is negative. For example,
as an object is lifted, the work done by the gravitational force on the object is
negative. The factor cos ! in the definition of W (Eq. 7.1) automatically takes care
of the sign.
If an applied force F is in the same direction as the displacement r, then Ɵ=0
and cos 0 = 1. In this case, Equation 7.1 gives
Work is a scalar quantity, and its units are force multiplied by length. Therefore,
the SI unit of work is the newton! meter (N· m). This combination of units is used so
frequently that it has been given a name of its own: the joule ( J).
An important consideration for a system approach to problems is to note that work
is an energy transfer. If W is the work done on a system and W is positive, energy is
transferred to the system; if W is negative, energy is transferred from the system. Thus, if
a system interacts with its environment, this interaction can be described as a transfer
of energy across the system boundary. This will result in a change in the energy stored
in the system. We will learn about the first type of energy storage in Section 7.5, after
we investigate more aspects of work.
7.3 The Scalar Product of Two Vectors
Because of the way the force and displacement vectors are combined in Equation 7.1,
it is helpful to use a convenient mathematical tool called the scalar product of two
vectors. We write this scalar product of vectors A and B as A.B. (Because of the dot
symbol, the scalar product is often called the dot product.)
In general, the scalar product of any two vectors A and B is a scalar quantity equal
to the product of the magnitudes of the two vectors and the cosine of the angle Ɵ between
Comparing this definition to Equation 7.1, we see that we can express Equation 7.1
as a scalar product:
7.4 Work Done by a Varying Force
Consider a particle being displaced along the x axis under the action of a force that varies with position. The particle is displaced in the direction of increasing x from to . In such a situation, we cannot use to calculate the work done by the force because this relationship applies only when F is constant in magnitude and direction. However, if we imagine that the particle undergoes a very small displacement , shown in Figure 7.7a, the x component Fx of the force is approximately constant over this small interval, for this small displacement, we can approximate the work done by the force as
If we imagine that the Fx versus x curve is divided into a large number of such intervals, the total work done for the displacement from xi to xf is approximately equal to the sum of a large number of such terms:
If the size of the displacements is allowed to approach zero, the number of terms in the sum increases without limit but the value of the sum approaches a definite value equal to the area bounded by the Fx curve and the x axis:
Therefore, we can express the work done by Fx as the particle moves from xi to xf as :
Work Done by a Spring
A model of a common physical system for which the force varies with position is shown in Figure 7.10. A block on a horizontal, frictionless surface is connected to a spring. If the spring is either stretched or compressed a small distance from its unstretched (equilibrium) configuration, it exerts on the block a force that can be expressed as :
Where x is the position of the block relative to its equilibrium (x = 0) position and k is a positive constant called the force constant or the spring constant of the spring. In other words, the force required to stretch or compress a spring is proportional to the amount of stretch or compression x. This force law for springs is known as Hooke’s law.
Because the spring force always acts toward the equilibrium position (x=0), it is sometimes called a restoring force. If the spring is compressed until the block is at the point –xmax and is then released, the block moves from - xmax through zero to + xmax. If the spring is instead stretched until the block is at the point + xmax and is then released, the block moves from + xmax through zero to - xmax. It then reverses direction, returns to + xmax, and continues oscillating back and forth. Suppose the block has been pushed to the left to a position - xmax and is then released. Let us identify the block as our system and calculate the work Ws done by the spring force on the block as the block moves from xi = - xmax to xf = 0. Applying equation 7.7 and assuming the block may be treated as a particle, we obtain
Therefore, the work done by this applied force (the external agent) on the block–spring system is
The work done by an applied force on a block–spring system between arbitrary positions
of the block is (7.12)
Notice that this is the negative of the work done by the spring as expressed by Equation 7.11. This is consistent with the fact that the spring force and the applied force are of equal magnitude but in opposite directions.
7.5 Kinetic Energy and the Work–Kinetic Energy Theorem
We have investigated work and identified it as a mechanism for transferring energy into a system. One of the possible outcomes of doing work on a system is that the system changes its speed. In this section, we investigate this situation and introduce our first type of energy that a system can possess, called kinetic energy.
Using Newton’s second law, we can substitute for the magnitude of the net force ma
and then perform the following chain-rule manipulations on the integrand:
In general, the kinetic energy K of a particle of mass m moving with a speed v is defined
Kinetic energy is a scalar quantity and has the same units as work. It is often convenient to write Equation 7.14 in the form
Another way to write this is Kf = Ki + 3W, which tells us that the final kinetic energy is equal to the initial kinetic energy plus the change due to the work done. Equation in this above is an important result known as the work–kinetic energy theorem:
“In the case in which work is done on a system and the only change in the system is in its speed, the work done by the net force equals the change in kinetic energy of the system.”
The work–kinetic energy theorem indicates that the speed of a particle will increase if
the net work done on it is positive, because the final kinetic energy will be greater than the initial kinetic energy. The speed will decrease if the net work is negative, because the final kinetic energy will be less than the initial kinetic energy.
Because we have only investigated translational motion through space so far, we arrived at the work–kinetic energy theorem by analyzing situations involving translational motion. Another type of motion is rotational motion, in which an object spins about an axis. We will study this type of motion in Chapter 10. The work–kinetic energy theorem is also valid for systems that undergo a change in the rotational speed due to work done on the system. The windmill in the chapter opening photograph is an example of work causing rotational motion.
7.6 The Nonisolated System—Conservation of Energy
We have seen examples in which an object, modeled as a particle, is acted on by various forces, resulting in a change in its kinetic energy. This very simple situation is the first example of the nonisolated system—a common scenario in physics problems. Physical problems for which this scenario is appropriate involve systems that interact with or are influenced by their environment, causing some kind of change in the system. If a system does not interact with its environment it is an isolated system, which we will study in Chapter 8.
The work–kinetic energy theorem is our first example of an energy equation appropriate for a nonisolated system. In the case of the work–kinetic energy theorem, the interaction is the work done by the external force, and the quantity in the system that changes is the kinetic energy.
We have seen only one way to transfer energy into a system so far—work. We mention
below a few other ways to transfer energy into or out of a system. The details of these processes will be studied in other sections of the book. We illustrate these in Figure 7.17 and summarize them as follows:
Work, as we have learned in this chapter, is a method of transferring energy to a system by applying a force to the system and causing a displacement of the point of application of the force.
Mechanical waves (Chapters 16–18) are a means of transferring energy by allowing a disturbance to propagate through air or another medium.
Heat (Chapter 20) is a mechanism of energy transfer that is driven by a temperature difference between two regions in space.
Matter transfer (Chapter 20) involves situations in which matter physically crosses the boundary of a system, carrying energy with it.
Electrical Transmission (Chapters 27–28) involves energy transfer by means of electric currents. This is how energy transfers into your hair dryer , stereo system, or any other electrical device.
Electromagnetic radiation (Chapter 34) refers to electromagnetic waves such as light, microwaves, radio waves, etc. Examples of this method of transfer include cooking a baked potato in your microwave oven and light energy traveling from the Sun to the Earth through space.
One of the central features of the energy approach is the notion that we can neither create nor destroy energy—energy is always conserved. Thus, if the total amount of energy in a system changes, it can only be due to the fact that energy has crossed the boundary of the system by a transfer mechanism such as one of the methods listed above. This is a general statement of the principle of conservation of energy. We can describe this idea mathematically as follows:
7.7 Situations Involving Kinetic Friction
A book sliding to the right on a horizontal surface slows down in the presence of a force of kinetic friction acting to the left. The initial velocity of the book is vi , and its final velocity is vf . The normal force and the gravitational force are not included in the diagram because they are perpendicular to the direction of motion and therefore do not influence the book’s speed.
The work–kinetic energy theorem is valid for a particle or an object that can be modeled as a particle. For these kinds of situations, Newton’s second law is stil valid for the system, even though the work–kinetic energy theorem is not.
Newton’s second law (x component only) by a displacement ∆x of the book:
we know that the following relationships
where vi is the speed at t = 0 and vf is the speed at time t .
The net force on the book is the kinetic friction force fk, which is directed opposite to the displacement ∆x . Thus
(x)Δx = -fx Δx = f2- i2 = ΔK
-fk Δx =Δ K
For a nonisolated system, we can equate the change in the total energy stored in the system to the sum of all the transfers of energy across the system boundary. For an isolated system, the total energy is constant—this is a statement of conservation of energy. If a friction force acts, the kinetic energy of the system is reduced and the appropriateequation to be applied is
The conclusion of this discussion is that the result of a friction force is to transform kinetic energy into internal energy, and the increase in internal energy is equal to the decrease in kinetic energy.
Quick Quis 7.11
You are traveling along a freeway at 65 mi/h. Your car has kinetic energy. You suddenly skid to a stop because of congestion in traffic. Where is the kinetic energy that your car once had?
(a) All of it is in internal energy in the road.
(b) All of it is in internal energy in the tires.
(c) Some of it has transformed to internal energy and some of it transferred away by mechanical waves.
(d) All of it is transferred away from your car by various mechanisms.
Answer : (c). The brakes and the roadway are warmer, so their internal energy has increased. In addition, the sound of the skid represents transfer of energy away by mechanical waves.
Power is the time rate of energy transfer or valid for any means of energy transfer. If an external force is applied to an object (which we assume acts as a particle), and if the work done by this force in the time interval ∆t is W, then the average power during this interval is defined as
In a manner similar to the way we approached the definition of velocity and acceleration,we define the instantaneous power P as the limiting value of the average power as ∆t = 0
P = = = F . = F . V
The SI unit of power is joules per second ( J/s), also called the watt (W)(after James Watt): 1 W = 1 J/s = 1 kg m2s3 . A unit of power in the U.S. customary system is the horsepower (hp):
1 hp = 746 W
A unit of energy (or work) can now be defined in terms of the unit of power. One kilowatt-hour (kWh) is the energy transferred in 1 h at the constant rate of 1 kW =1 000 J/s. The amount of energy represented by 1 kWh is
1 kWH = (103W) (3600s) = 3.6 x 106 J
Quick quis 7.12
An older model car accelerates from rest to speed v in 10 seconds. A newer, more powerful sports car accelerates from rest to 2v in the same time period. What is the ratio of the power of the newer car to that of the older car?
(a) 0.25 (b) 0.5 (c) 1 (d) 2 (e) 4
Answer : (e). Because the speed is doubled, the kinetic energy is four times as large. This kinetic energy was attained for the newer car in the same time interval as the smaller kinetic energy for the older car, so the power is four times as large.
7.9 Energy and Automobile
Automobiles powered by gasoline engines are very inefficient machines. About 67% of the energy available from the fuel is lost in the engine. Approximately 10% of the available energy is lost to friction in the transmission, drive shaft, wheel and axle bearings, and differential. Friction in other moving parts transforms approximately 6% of the energy to internal energy, and 4% of the energy is used to operate fuel and oil pumps and such accessories as power steering and air conditioning. This leaves a mere 13% of the available energy to propel the automobile.
For large objects, the resistive force fa associated with air friction is proportional to the square of the speed
Fa = DρAv 2
where D is the drag coefficient, / is the density of air, and A is the cross-sectional area of the moving object.