Page 15 - Fisika Terapan for Engineers and Scientists

7.3 Kinetic Energy 215

dv dv dx dv dv
v v (7.20)
dt dx dt dx dx
Consequently, the work becomes
x 2 dv x 2 dv v 2 1 2 v 2
m dx m v dx m v dv m v `
2
dt dx
(7.21)
x 1 x 1 v 1 v 1
1
2
1
mv mv 2
2 2 2 1
or
1
2
1
W mv mv 2 (7.22)
2 2 2 1
This shows that the change in the square of the speed is proportional to the work done
by the force.
Although we have here obtained the result (7.22) for the simple case of motion
along a straight line, it can be shown that the same result is valid for motion along a
curve, in three dimensions.
According to Eq. (7.22), whenever we perform positive work on the particle, we
2
1
increase the “amount of mv ” in the particle; and whenever we perform negative
2
work on the particle (that is, when we let the particle perform work on us), we decrease
2
2
1
1
the “amount of mv ” in the particle. Thus, the quantity mv is the amount of work
2
2
stored in the particle, or the kinetic energy of the particle. We represent the kinetic energy
by the symbol K:
1
K mv 2 (7.23) kinetic energy
2
With this notation, Eq. (7.22) states that the change of kinetic energy equals the net work
done on the particle; that is,
K K W (7.24)
2 1
or

K W (7.25) work–energy theorem

This result is called the work–energy theorem. Keep in mind that the work in Eqs.
(7.22), (7.24), and (7.25) must be evaluated with the net force; that is, all the forces
that do work on the particle must be included in the calculation.
When a force does positive work on a particle initially at rest, the kinetic Water has a small Gravity does work
kinetic energy. on water…
energy of the particle increases. The particle then has a capacity to do work:
if the moving particle subsequently is allowed to push against some obstacle, then …which gains a
this obstacle does negative work on the particle and simultaneously the particle does large kinetic energy.
positive work on the obstacle.When the particle does work, its kinetic energy decreases.
The total amount of work the particle can deliver to the obstacle is equal to its kinetic
energy.Thus, the kinetic energy represents the capacity of a particle to do work by virtue of
its speed.
The acquisition of kinetic energy through work and the subsequent production of
work by this kinetic energy are neatly illustrated in the operation of a waterwheel Water does work
on wheel, losing
driven by falling water. In a flour mill of an old Spanish Colonial design, the water FIGURE 7.16 kinetic energy.
runs down from a reservoir in a steep, open channel (see Fig. 7.16).The motion of the Water pushing on a
water particles is essentially that of particles sliding down an inclined plane. If we horizontal waterwheel.