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310 Chapter 7 Conservation of Energy and an Introduction to Energy and Work

Reflect h = 8.6 cm
= 8.6 cm
h h
The height seems pretty reasonable, 8.6 cm. But in the work-energy approach
Thinking about conservation of energy, because we don’t have to worry about
the tension force does no work, the maximum the tension force at all because it v v v = 1.3 m/s
height reached should be the same as if the spider does no work on the spider. i i
had initially been moving straight up at 1.3 m/s If the spider was initially
h h
= 8.6 cm
without being attached to the silk, and we can moving straight up: h = 8.6 cm
check that. In both cases the gravitational force 1 1
2
does the same amount of (negative) work to W net = mv − mv i 2 v v v = 1.3 m/s
i i
f
reduce the spider’s kinetic energy to zero, because 2 2
the tension does no work. Because the tension is W grav = g F h cos 180° =− mgh
always perpendicular to the motion, we know it W W mgh = 0 − 1 mv
2
can cause only the direction of motion to change, net = grav = − 2 i
not the speed. v 2 (1.3 m/s) 2
The tension force in this problem is h = i = 2
complicated because its magnitude and direction 2g 2 × 9.8 m/s
change as the spider moves through its swing. h = 0.086 m
NOW WORK Problems 3 and 7 from The Takeaway 7-5.

Example 7-6 illustrates an important point: If a force is always exerted perpendic-

ular to an object’s path, it does zero work on the object. This enabled us to ignore the
effects of the tension force in this problem. We’ll use this same idea in many contexts!
The Work-Energy Theorem Is Not Just for Objects
When you look at the chapter-opening sequence of photographs, you will see, given
our discussion in Section 1-3 that we cannot treat the ball as an object during this
,
time; we must use a system description. Additionally, we discussed the need to describe
the motion of one point in a system to represent the whole system if we want to use the
object model. For the object model, we normally choose this point to be the center of
mass. In Section 4-4 we discussed how to qualitatively locate the center of mass, which
will be sufficient for what we are doing now. (In Section 9-7 we will learn how to cal-
culate the location of the center of mass, when it becomes clearer in the context we will
be studying.) Examining the chapter-opening sequence of photographs, you see the ball
expand as it moves away from the wall. Because it was at rest when it was completely
against the wall, it is speeding up in this process. You can imagine what this sequence
of photographs would look like if it were longer and involved the ball initially striking
the wall. The ball would compress as it slowed to a stop.
Given what we have been discussing so far in this chapter, our first reaction might
be to think the wall has done work on the ball. After all, the ball’s kinetic energy has to
go to zero at the instant it is fully compressed against the wall, as it changes direction.
The wall does no work on the ball, however. Our definition of work says that the force
must be exerted on the object as the point of contact of the force on the object moves
for there to be work done on the object. In this example, the point of contact of the
ball and the wall does not move, so the force exerted by the wall does not do work.
Actually, this makes a lot of sense if you think about it: You can lean against a wall all
day, and it is not going to make you start moving!
The shortcut we took in deriving the work-energy theorem for an object ( Equa-
tion 7-9 ) is clear in its name: We used the object model. By definition, for the object
model to be valid, the center of mass and any point on the object must move the same
distance. That is not true for a system such as a compressible ball. In the chapter-
opening photographs, we see that the point of contact for the force does not move,
but using our qualitative understanding of the center of mass, we see the center of
mass clearly does get further from the wall as the ball expands. Work is determined
by the displacement of the point of contact while the force doing the work is being
exerted, but Equation 7-9 is derived from an equation that describes the motion of
the center of mass. This means the displacement of the center of mass while the force
is being exerted describes change in the kinetic energy of the system. For a system,

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