Page 23 - Fisika Terapan for Engineers and Scientists

7.4 Gravitational Potential Energy 223

(a) (b)
y

Initially, energy
P 1
is all potential.

At the bottom,
y 1 energy is all kinetic.

P 2
O x

FIGURE 7.23 (a) A roller coaster. (b) Profile
of a roller coaster. The roller-coaster car descends
from P to P .
2
1

which gives
2
v 22 9.81 m/s 38 m 27m/s
2
Note that according to Eq. (7.41) the final velocity is independent of the mass of
the car; since both the kinetic energy and the gravitational potential energy are
proportional to mass, the mass cancels in this calculation.

COMMENT: This example illustrates how energy conservation can be exploited
to answer a question about motion. To obtain the final speed by direct computa-
tion of forces and accelerations would have been extremely difficult—it would have
required detailed knowledge of the shape of the path down the hill. With the Law
of Conservation of Energy we can bypass these complications.

ENERGY CONSERVATION IN ANALYSIS
PROBLEM-SOLVING TECHNIQUES
OF MOTION

As illustrated by the preceding example,the use of energy con- y coordinate. However, the change in the potential energy
servation in a problem of motion typically involves three steps: does not depend on the choice of this level, and therefore any
choice will lead to the same result for the change of kinetic
1 First write an expression for the energy at one point of
energy. Thus, you can make any choice of zero level, but
the motion [Eq. (7.38)].
you must continue to use this choice throughout the entire
2 Then write an expression for the energy at another point calculation. You will usually find it convenient to place the
[Eq. (7.39)]. zero level for the y coordinate either at the final position of
3 And then rely on energy conservation to equate the two the particle (as in the preceding example), or at the initial
expressions [Eq. (7.40)].This yields one equation, which position, or at some other distinctive height, such as the
can be solved for the unknown final speed or the unknown bottom of a hill or the ground floor of a building. And always
final position (if the final speed is known). remember that the formula U mgy for the gravitational
potential energy assumes that the y axis is directed verti-
Note that the value of the gravitational potential energy
cally upward.
U mgy depends on the level from which you measure the