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7-6 Potential energy is energy related to reversible changes in a system’s configuration 315

represented by (1 − ) and (x − 1) where 1 F and 2 F are Skills in Action

F
,
F

x

2
1
constants.

(a) On a single graph construct a representation of both 8. An object with a mass of 0.5 kg and a velocity of 10 m/s
forces as a function of position. enters a region in which a changing force F in the x direction
(b) Refine the representation by shading the area that is exerted on the object. The object moves from an initial

represents the work done on the object by these forces position = 0.x

from =x 0 to = 2m. (a) Analyze the data in the table to express F / m as a

x
(c) Apply mathematical reasoning to support the claim that function of x .
the net work done on the object as the object moves x (m) 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8

from =x 0 to = 2m is zero for any values of 1 F and . 2 F

x

EX 7. Before giving a slick ice cube at the bottom of a large, F / m −10 −9 −8 −7 −6 −5 −4 −3 −2 −1
7-6 (N/kg)
smooth ceramic bowl a flick with your finger, you want

to calculate the height to which the flick will cause the (b) Calculate the initial kinetic energy of the object.

ice cube to rise. You estimate the mass of the ice cube to (c) Predict the point, , s x where the object momentarily
be between 25 and 50 g and the initial velocity of the ice comes to a stop and begins to move in the negative
cube after it leaves your finger at the bottom of the bowl x direction. Hint : Apply the work-energy theorem
to be between 1 and 2 m/s. to find the point where the object stops (the kinetic
(a) Justify the selection of the use of the energy conser- energy of the object is zero).

vation principle rather than Newton’s second law to (d) Justify the claim that the kinetic energy of the object
make this prediction. will have the value calculated in part (b) when, after

(b) Construct a free-body diagram of the ice cube at some stopping, it returns to the position = 0.x

point as it rises along the inner surface of the bowl 9. An ideal spring is used to stop blocks as they slide along a
and annotate the diagram with a vector representing track with negligible friction, as shown in the figure. Mea-
the displacement of the ice cube at that instant. surements are made of the maximum displacement of the

(c) Using the diagram constructed in part (b), explain spring when struck by a 0.20-kg block for different velocities.
why the normal force exerted by the wall of the bowl Use these data to determine the spring constant graphically.
on the ice cube does not do work on the ice cube. Velocity Displacement
(d) Let a vector along a short segment of the path of the (m/s) (cm)

ice cube as it rises in the bowl be d Construct a dia-
.

gram in which the displacement d is the hypotenuse 0.8 3.5

of a triangle with one component that is antiparallel 1.8 7.2
to the gravitational force. In the diagram, label the 2.8 11
angle between the displacement and this component 3.7 14.5
as θ. Using this diagram explain why the sum of dis- 4.3 17.5

placements along the field is equal to the height the
ice cube goes up the bowl. 5.3 22

(e) Using the diagram constructed in part (d) explain how
the height to which the ice cube rises can be predicted. v v v
(f) Explain why the predicted height is insensitive to your

uncertainty in the mass of the ice cube but very sensi-
tive to your uncertainty in the initial velocity.
Potential energy is energy related to reversible
7-6
changes in a system’s configuration
When we do work on a spring to compress it, we don’t give it kinetic energy, but we
could use that compressed spring to give kinetic energy to a ball, for instance, by using
the spring as a launcher. Let’s go back to our photographs at the beginning of the chapter.
When the ball comes to rest compressed against the wall some of its kinetic energy gets
stored in the compression of the ball, just like the energy stored in springs as we saw
in the last section. The more elastic the ball is, the greater the proportion of the kinetic
energy that gets converted into energy stored in the compression of the ball. Some of
the energy goes into warming up the ball, and some into making noise. The energy that
.
goes into these types is said to be dissipated Dissipated energy is energy that has been
converted into a type other than kinetic or potential energy, that cannot ever be fully
Uncorrected proofs have been used in this sample. Copyright © Bedford, Freeman & Worth Publishers.
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