05-02-2013, 12:19 PM
Work and Kinetic Energy
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INTRODUCTION
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.
In this chapter, we first introduce the concept of work. Work is done by a force
acting on an object when the point of application of that force moves through
some distance and the force has a component along the line of motion. Next, we
define kinetic energy, which is energy an object possesses because of its motion. In
general, we can think of energy as the capacity that an object has for performing
work. We shall see that the concepts of work and kinetic energy can be applied to
the dynamics of a mechanical system without resorting to Newton’s laws. In a complex
situation, in fact, the “energy approach” can often allow a much simpler
analysis than the direct application of Newton’s second law. However, it is important
to note that the work–energy concepts are based on Newton’s laws and therefore
allow us to make predictions that are always in agreement with these laws.
This alternative method of describing motion is especially useful when the
force acting on a particle varies with the position of the particle. In this case, the acceleration
is not constant, and we cannot apply the kinematic equations developed
in Chapter 2. Often, a particle in nature is subject to a force that varies with the position
of the particle. Such forces include the gravitational force and the force exerted
on an object attached to a spring. Although we could analyze situations like
these by applying numerical methods such as those discussed in Section 6.5, utilizing
the ideas of work and energy is often much simpler. We describe techniques for
treating complicated systems with the help of an extremely important theorem
called the work–kinetic energy theorem, which is the central topic of this chapter.
WORK DONE BY A CONSTANT FORCE
Almost all the terms we have used thus far—velocity, acceleration, force, and so
on—convey nearly the same 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. That new term is work.
WORK DONE BY A VARYING FORCE
Consider a particle being displaced along the x axis under the action of a varying
force. The particle is displaced in the direction of increasing x from x xi to x
xf . In such a situation, we cannot use W (F cos )d 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
x, shown in Figure 7.7a, then the x component of the force Fx is approximately
constant over this interval; for this small displacemen.