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330 Chapter 7 Conservation of Energy and an Introduction to Energy and Work

to your face and releasing it from rest, as shown in 25. As you pedal a bike up a hill, predict
part (a) of the figure. (a) the sign of the work done by the gravitational

force on you,
(a) (b) (b) the sign of the work done by your foot on the bike

pedal as it pushes the pedal down, and
(c) the sign of the work done by the pedal on your
foot as the pedal moves downward.
L L
0
Hint : Remember that predictions are always supported
∆h by reasoning and that coordinate systems are chosen
∆L for convenience.

26. A 1.0-kg object is at rest when a force is exerted on
the object. The positions at several times are shown in
the figure.
t = 0.5 s
t = 0 s t = 1 s t = 1.5 s t = 2 s
0.0 0.2 0.4 0.6 0.8
Note: Figure not drawn to scale. x (m)

(a) Use mathematical reasoning to express displace-

(a) Predict what will happen to the relative height, ∆h , ment as a function of time for the object by evalu-
to which the ball rises if the ball is pushed rather ating the acceleration.
than released from rest in terms of the work done, (b) Calculate the numerical value of the force exerted

W on ball , by the push. on the object.
A teacher tires of this demonstration and attaches (c) Express work done by the force on the object and
the bowling ball to an ideal spring with constant k . kinetic energy of the object as functions of time.
As shown in part (b) of the figure, the equilibrium
length of the spring hanging vertically when the The external force is switched off and the object is
L

ball is attached is . 0 Lying on the floor with the returned to the original initial position. At the instant
bowling ball suspended at rest above his chest the force is switched on again a second object with

he pulls down slightly on the ball a distance ∆L , a mass of 2.0 kg enters along a parallel path with an

v
and releases it. initial position =x 0 and an initial speed of = 1m/s.

(b) Take the ball to be initially at rest above the teacher’s Interactions between the two objects are negligible.
.

L +∆L What are the At = 1t s the external force is switched off.
chest with a spring length 0

forces exerted on the ball at this time? (d) Express the displacements of both objects as a

(c) Using the conservation of energy, describe the function of time.
motion of the ball after the teacher releases it. (e) Express the kinetic energy of the two-object system

(d) Which demonstration is safer, do you think? Justify as a function of time.

your answer in terms of the subsequent motion of (f) Explain why the kinetic energy of the two-object
the ball in each case, (a) and (c). system is constant for > 1s.t
Prep for the AP ® Exam
AP Group Work
®
Directions: The following problem is designed to be done as and she draws the graph of force versus spring compres-
group work in class. sion as shown in the following graph. Gwen argues that
As part of a lab experiment that uses a horizontal air track, the energy added to the spring is the area under the force-

Allison pushes an air-track cart of mass m up against a spring displacement graph, and points out that the two shaded
to compress the spring with spring constant k by an amount triangles have the same area. So a second compression by a

∆x from its equilibrium length. The air track has negligible distance d beyond a first compression by a distance d doubles

friction. When Allison lets go, the spring launches the cart. the area and therefore doubles the energy. The energy is pro-
Allison’s lab partner Gwen predicts that because the portional to the velocity squared. So, Gwen argues, the square
force increases linearly as the spring is compressed, the of the velocity should be proportional to the compression.
square of the velocity when the spring is released should
increase in proportion to the compressed length. So the F
velocity should increase in proportion to the square root
of the displacement. To support her claim she refers to the
kinematic equation kd kd
d d x
v 2 = v 2 + 2a ∆x
f i
Uncorrected proofs have been used in this sample. Copyright © Bedford, Freeman & Worth Publishers.
Distributed by Bedford, Freeman & Worth Publishers. For review purposes only. Not for redistribution.
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