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7-6 Potential energy is energy related to reversible changes in a system’s configuration 323

analysis of the problem. Represent each characteristic (d) Apply the principle of energy conservation and evalu-
with a label that will be used in an analysis. ate the distance that the block moves down the incline

(b) Construct a free-body diagram for the box. from its release point to the stopping point.

(c) Construct a diagram representing the decomposition (e) Analyze the dynamics of the block using an alterna-
of the gravitational force into components parallel tive choice of system to that used in your analysis in
and perpendicular to the ramp. One of the angles in part (d).
your vector diagram will be the same as the angle 6. Ben says to Jerry that the potential energy change of the

of inclination of the ramp. Mark this angle on your Ben–Earth system as he climbs a ladder from the first
vector diagram. Add the displacement to this dia- rung to the fourth rung is smaller than the potential
gram and mark the angle, taken to be positive in the energy change of the same system as he climbs from the
counter clockwise direction, between the displacement second rung to the fifth rung because gravitational poten-
and the gravitational force, to your diagram. tial energy is proportional to height and the height of the

(d) Express the components of the gravitational force fifth rung is greater than the height of the fourth rung. In
along the ramp and perpendicular to the ramp in response, Jerry draws a diagram with two ladders with
terms of g F and θ. one in the basement and one on the first floor. Construct

(e) Express the work done on the box by the gravita- Jerry’s diagram and annotate the diagram to support his
tional force in terms of θ. Hint : To be sure of your evaluation of Ben’s claim.

expression you can use the trigonometric identity 7. Carrie’s group is analyzing the motion of a cart that

moves up a ramp. Carrie claims that when the cart slows
α
α
)
)
cos( α − β = cos() cos( β + sin() sin( β)
down the kinetic energy is negative. She justifies this claim
to express the work done on the box by the gravita- by stating that “the work-energy theorem for an object
tional force. says that when a force pushes an object the kinetic energy

(f) Apply the object model to the box and express the final increases but now the force is pulling the object.” Andrea
kinetic energy of the box in terms of the work done on disagrees and claims that the kinetic energy of the cart is
the box by the external force and the gravitational field. never negative anywhere on the ramp but doesn’t provide

(g) Consider the Earth–box system and apply energy con- reasoning to support her claim.

servation to express the final kinetic energy in terms of (a) Treating the cart as an object upon which Earth exerts
the work done by the external force and the change in a gravitational force, justify Andrea’s claim.

gravitational potential energy of the Earth–box system. (b) Carrie responds by claiming that by treating the cart–
(h) Compare the expressions from parts (f) and (g) and Earth pair as an isolated system, no work is done and

explain similarities and differences. so according to the work-energy theorem the kinetic
(i) Calculate the final speed of the box. energy of the system cannot change. And because the
EX 5. A 12.0-kg block ( M ) is released from rest on an incline total kinetic energy of the system is the sum of the
7-8
that makes an angle of 28.0 ° with the horizontal. Below kinetic energies of the components, for the sum to be

the block is a spring that has a spring constant of zero one of the terms in this sum must be negative.
13,500 N/m. The figure shows the initial arrangement. The Evaluate Carrie’s explanation of why kinetic energy
friction between the block and incline is negligible. The can be negative.
block momentarily stops when it compresses the spring by
5.50 cm. You will analyze the dynamics of the block with Skills in Action
the goal of predicting the displacement of the block from

its initial position at the instant that it momentarily stops 8. Several identical springs, with spring constant k , are
while compressing the spring. attached end-to-end, as shown in the first diagram below.
An external force is exerted on the first spring with the
response shown in the second diagram. The springs are
M
stretched as shown.
k k k k k
q
(a) Construct a diagram for your analysis of the scenario ∆x

that includes initial (i) and final (f) states, displace- ∆x 1 total
ment prior to collision with the spring, displacement F external,x
after collision with the spring, and the angle of incli-
nation of the ramp with labels for each variable to be
used in your analysis.
(b) Justify your selection of the system to be analyzed. (a) Justify the claim that

(c) Express potential and kinetic energies at the initial

and final states mathematically. If work is done to ∆ total = n F external,x
x
your system express the work mathematically. k
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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