pendulum – Earth system is transferred back and forth between kinetic energy K
and gravitational potential energy U, with the sum KUbeing constant. If we
know the gravitational potential energy when the pendulum bob is at its highest
point (Fig. 8-7c), Eq. 8-17 gives us the kinetic energy of the bob at the lowest
point (Fig. 8-7e).
For example, let us choose the lowest point as the reference point, with the
gravitational potential energy U 2 0. Suppose then that the potential energy at
the highest point is U 1 20 J relative to the reference point. Because the bob mo-
mentarily stops at its highest point, the kinetic energy there is K 1 0. Putting these
values into Eq. 8-17 gives us the kinetic energy K 2 at the lowest point:
K 2 0 0 20 J or K 2 20 J.
Note that we get this result without considering the motion between the highest
and lowest points (such as in Fig. 8-7d) and without finding the work done by any
forces involved in the motion.186 CHAPTER 8 POTENTIAL ENERGY AND CONSERVATION OF ENERGY
Checkpoint 3
The figure shows four
situations — one in
which an initially sta-
tionary block is dropped
and three in which the
block is allowed to slide
down frictionless ramps.
(a) Rank the situations
according to the kinetic energy of the block at point B, greatest first. (b) Rank them
according to the speed of the block at point B, greatest first.AB B B B( 1 )( 2 )( 3 )( 4 )System:Because the only force doing work on the child
is the gravitational force, we choose the child – Earth system
as our system, which we can take to be isolated.
Thus, we have only a conservative force doing work in
an isolated system, so we canuse the principle of conserva-
tion of mechanical energy.
Calculations:Let the mechanical energy be Emec,twhen the
child is at the top of the slide and Emec,bwhen she is at the
bottom. Then the conservation principle tells us
Emec,bEmec,t. (8-19)Sample Problem 8.03 Conservation of mechanical energy, water slide
The huge advantage of using the conservation of energy in-
stead of Newton’s laws of motion is that we can jump from
the initial state to the final state without considering all the
intermediate motion. Here is an example. In Fig.8-8, a child
of mass mis released from rest at the top of a water slide,
at height h8.5 m above the bottom of the slide.
Assuming that the slide is frictionless because of the water
on it, find the child’s speed at the bottom of the slide.
KEY IDEAS
(1) We cannot find her speed at the bottom by using her ac-
celeration along the slide as we might have in earlier chap-
ters because we do not know the slope (angle) of the slide.
However, because that speed is related to her kinetic en-
ergy, perhaps we can use the principle of conservation of
mechanical energy to get the speed. Then we would not
need to know the slope. (2) Mechanical energy is conserved
in a system ifthe system is isolated and ifonly conservative
forces cause energy transfers within it. Let’s check.
Forces:Two forces act on the child. The gravitational
force,a conservative force, does work on her. The normal
forceon her from the slide does no work because its direc-
tion at any point during the descent is always perpendicular
to the direction in which the child moves.
Figure 8-8A child slides down a water slide as she descends a
heighth.hThe total mechanical
energy at the top
is equal to the total
at the bottom.