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

WATCH OUT ! Table 7-1 summarizes the relationship between the angle at which a force is exerted
on an object relative to its displacement and the work done on the object. It only con-
Don’t let the signs fool you! siders an angle between 0 and 180°.
Work is a scalar quantity; that
is, it has no direction. Work TABLE 7-1 Work Done when the Force Exerted on an Object Is Constant
is the change in energy of an
object or a system due to a force If the angle between a force F exerted on an object

exerted on it while it moves. and the displacement d of the object is… …then the work done on the object is…
The sign of work determines 0° to 90 ° positive
if the work is increasing (+) or
decreasing (−) the energy of the 90° zero
object or system. Add work 90° to 180° negative
terms to find the total change in
energy of the object or system.
Calculating Work Done when Multiple Forces
Are Exerted on an Object

What if more than one force is exerted on an object as it moves? Then the total work
AP ® Exam Tip done on the object is the sum of the work done on the object by each object or system
It is important to be able exerting a force on it individually. We simplify this language by saying it is the work
to distinguish conceptually done by the forces, but remember that the energy has to come from whatever is exert-
between when a particular force ing the force. For example, four forces are exerted on the screen in Figure 7-4a: the ten-
is doing positive or negative sion in the rope, the gravitational force, the normal force, and the force of friction. The

work. Note that an object can free-body diagram in Figure 7-7 shows all of these forces, as well as the displacement .d

still be speeding up while a Note that we draw d to one side in the diagram to avoid confusing it with the forces,
particular force is doing negative and that the direction of motion is in the positive x direction as indicated by the
work on it; in which case, there dashed arrow. Because there are several forces exerted on the object, add subscripts to
must be another force doing your angles so you can tell which angle goes with which force.
positive work on the object. We use Equation 7-2 to determine the work that each force does on the screen:

F
Tension force: This force T (which we labeled F in Figure 7-4a) is exerted at an
F n F angle θ T with respect to the direction of motion, so W tension = F d cos θ T .
T

q T T Gravitational force: The gravitational force g F is perpendicular to the direction of
F k x motion, so the angle θ in Equation 7-2 is 90° for this force. Because cos 90 °= 0,
the work done by the gravitational force is W = 0.
d grav
Normal force: The normal force n F is also perpendicular to the direction of motion.
So like the gravitational force, the normal force does no work: W normal = 0.

F g Kinetic friction force: Because friction opposes sliding, the kinetic friction force F points
k
Always draw the displacement d in the direction opposite to the motion. The angle to use in Equation 7-2 is therefore
to one side so you don’t confuse 180°, and W = F d cos 180 °= −Fd. When a force is exerted to oppose the
it with the forces. friction k k
motion of the object, the value of the work done by that force is negative.
Figure 7-7 Calculating the work The total work done on the screen by all four forces is the sum of the work done
done by multiple forces The free- by each force exerted on the object:
body diagram for the screen shown
in Figure 7-4a. To find the work done W total = W tension + W grav + W normal + W friction
by each force exerted on the object, = Fd cos θ + Fd cos (90) cos(90 °+ −Fd )
)(

°+ Fd
draw the displacement d on the T T g n k
F
diagram, and label angles to match = Fd cos θ − Fd = ( T cos θ − F k )d
T
T
T
k
forces.
F
If the horizontal component T cos θ T of the tension force (which does positive work)
F (which does negative work), the
is greater in magnitude than the kinetic friction force k
WATCH OUT ! total work done on the screen is positive and the screen speeds up. But if the magnitude
cos
F
θ T is less than F , the total work done on the screen is negative and the screen
of T
k
Forces are not what do work! slows down. Note that the kinetic friction force cannot do any work on an object unless
Forces are just a way to describe that object is already in motion, or the surface is in motion, or some other force is causing
the surface or object to move. This makes sense: Because the ground doesn’t move, you
an interaction. When we talk can stand on the ground all day and if you don’t try to move, it isn’t going to make you.
about a force doing work, we While the dog lying on the screen in Figure 7-5 would not do any work itself, the
are using a sort of abbreviation
for the work done by whatever presence of the dog would increase the normal force and therefore the friction force, and
is exerting the force. so the negative work done by the friction force. So the tension force needed to drag the
screen increases, making the person do more work to pull the screen the same distance.
Uncorrected proofs have been used in this sample. Copyright © Bedford, Freeman & Worth Publishers.
Distributed by Bedford, Freeman & Worth Publishers. For review purposes only. Not for redistribution.
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