9781118230725.pdf

We can use the definition of the scaler (dot) product (Eq. 3-20) to write

(work done by a constant force), (7-8)

whereFis the magnitude of (You may wish to review the discussion of scaler
products in Module 3-3.) Equation 7-8 is especially useful for calculating the
work when and are given in unit-vector notation.
Cautions. There are two restrictions to using Eqs. 7-6 through 7-8 to calculate
work done on an object by a force. First, the force must be a constant force;that
is, it must not change in magnitude or direction as the object moves. (Later, we
shall discuss what to do with a variable forcethat changes in magnitude.) Second,
the object must be particle-like.This means that the object must be rigid;all parts
of it must move together, in the same direction. In this chapter we consider only
particle-like objects, such as the bed and its occupant being pushed in Fig. 7-3.
Signs for Work.The work done on an object by a force can be either positive
work or negative work. For example, if angle fin Eq. 7-7 is less than 90, then cos fis
positive and thus so is the work. However, if fis greater than 90(up to 180), then
cosfis negative and thus so is the work. (Can you see that the work is zero when
f 90 ?) These results lead to a simple rule. To find the sign of the work done by a
force, consider the force vector component that is parallel to the displacement:

d

:

F

:

F

:

.

WF

:

d

:

7-2 WORK AND KINETIC ENERGY 153

Figure 7-3A contestant in a bed race. We

can approximate the bed and its occupant

as being a particle for the purpose of cal-

culating the work done on them by the

force applied by the contestant.

F

A force does positive work when it has a vector component in the same direction

as the displacement, and it does negative work when it has a vector component in

the opposite direction. It does zero work when it has no such vector component.

Units for Work.Work has the SI unit of the joule, the same as kinetic energy.
However, from Eqs. 7-6 and 7-7 we can see that an equivalent unit is the newton-
meter (Nm). The corresponding unit in the British system is the foot-pound
(ftlb). Extending Eq. 7-2, we have

1J1kgm^2 /s^2 1Nm0.738 ftlb. (7-9)
Net Work. When two or more forces act on an object, the net workdone on
the object is the sum of the works done by the individual forces. We can
calculate the net work in two ways. (1) We can find the work done by each force
and then sum those works. (2) Alternatively, we can first find the net force
of those forces. Then we can use Eq. 7-7, substituting the magnitude FnetforF
and also the angle between the directions of and forf. Similarly, we can
use Eq. 7-8 with substituted for

Work–Kinetic Energy Theorem

Equation 7-5 relates the change in kinetic energy of the bead (from an initial
Ki 21 mv 02 to a later Kf^12 mv^2 ) to the work W(Fxd) done on the bead. For

F

:

F.

:

net

d

:

F

:

net

F

:

net

such particle-like objects, we can generalize that equation. Let Kbe the change
in the kinetic energy of the object, and let Wbe the net work done on it. Then

KKfKiW, (7-10)

which says that

We can also write

KfKiW, (7-11)

which says that

.

kinetic energy after

the net work is done

kinetic energy

before the net work

the net

work done



change in the kinetic

energy of a particle

net work done on

the particle .