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8.1 Potential Energy of a Conservative Force 243

2
v 3 2gh 3 2 9.81 m/s 330 m 80 m/s
(b) For a closed pipe with a small hole, the motion of a parcel reservoir
of water from the top of the upper reservoir to the hole at the bottom
of the pipe is complicated and unknown. However, we can find the
intake
final speed of the water by relying on the law of energy conservation
as applied to the system consisting of the entire volume of water
in the reservoir and the pipe. For this purpose, we must examine the
kinetic and the potential energy of the water.The water spurting out discharge
at the bottom has a large kinetic energy but a low potential energy.
In contrast, the water at the top of the upper reservoir has a high
potential energy, but next to no kinetic energy (while the water power plant chamber
spurts out at the bottom, the water level in the reservoir gradually
FIGURE 8.6 Cross-sectional view of hydroelectric
decreases; but the speed of this downward motion of the water level pumped-storage power plant.
is very small if the reservoir is large, and this speed can be ignored
compared with the large speed of the spurting water).
Consider, then, the energy changes that occur when a mass m of water, say,
1 kg of water, spurts out at the bottom of the pipe while, simultaneously, the water
level of the upper reservoir decreases slightly. As concerns the energy balance, this
effectively amounts to the removal of the potential energy of 1 kg from the top of
the reservoir and the addition of the kinetic energy of 1 kg at the bottom of the pipe.
All the water at intermediate locations, in the pipe and the reservoir, has the same
energy it had before. Thus, energy conservation demands that the kinetic energy
of the mass m of water emerging at the bottom be equal to the potential energy
of a mass m at the top:
1 2
2 mv mgh
This again gives

v 3 2gh 80 m/s

that is, the same result as in part (a).
COMMENT: Note that the way the water acquires the final speed of 80 m/s in the
cases (a) and (b) is quite different. In case (a), the water accelerates down the pipe
with the uniform free-fall acceleration g. In case (b), the water flows down the pipe
at a slow and nearly constant speed, and accelerates (strongly) only at the last
moment, as it approaches the hole at the bottom. However, energy conservation
demands that the result for the final speed of the emerging water be the same in
both cases.

✔ Checkup 8.1

QUESTION 1: The potential energy corresponding to the spring force F kx is
1
2
U 1 2 kx . Suppose that some new kind of force has a potential energy U kx 2 .
2
How does this new kind of force differ from the spring force?
QUESTION 2: A particle moves along the positive x axis under the influence of a con-
servative force. Suppose that the potential energy of this force is as shown in Fig. 8.5a.
Is the force directed along the positive x direction or the negative x direction?