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Introductory mechanics
Chapter 06 - Work And Kinetic Energy
Due at 11:59pm on Tuesday, October 14, 2008
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The Work-Energy Theorem
Learning Goal:
To understand the meaning and possible applications of the work-energy theorem.
In this problem, you will use your prior knowledge to derive one of the most important relationships in mechanics: the work-
energy theorem. We will start with a special case: a particle of mass
moving in the
x
direction at constant acceleration
.
During a certain interval of time, the particle accelerates from
to
, undergoing displacement
given by
.
Part A
Find the acceleration
of the particle.
Hint A.1
Some helpful relationships from kinematics
By definition,
.
Furthermore, the average speed is
,
and the displacement is
.
Combine these relationships to eliminate
.
Express the acceleration in terms of
,
, and
.
ANSWER:
=
Part B
Find the net force
acting on the particle.
Hint B.1
Using Newton's laws
Hint not displayed
Express your answer in terms of
and
.
ANSWER:
=
Part C
Find the net work
done on the particle by the external forces during the particle's motion.
Express your answer in terms of
and
.
ANSWER:
=
Part D
Substitute for
from Part B in the expression for work from Part C. Then substitute for
from the relation in Part A. This
will yield an expression for the net work
done on the particle by the external forces during the particle's motion in terms of
mass and the initial and final velocities. Give an expression for the work
in terms of those quantities.
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10/13/08 12:30 AM
MasteringPhysics
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http://session.masteringphysics.com/myct
Express your answer in terms of
,
, and
.
ANSWER:
=
The expression that you obtained can be rearranged as
The quantity
has the same units as work. It is called the
kinetic energy
of the moving particle and is denoted by
. Therefore, we can write
and
.
Note that like momentum, kinetic energy depends on both the mass and the velocity of the moving object. However, the
mathematical expressions for momentum and kinetic energy are different. Also, unlike momentum, kinetic energy is a
scalar. That is, it does not depend on the sign (therefore direction) of the velocities.
Part E
Find the net work
done on the particle by the external forces during the motion of the particle in terms of the initial and final
kinetic energies.
Express your answer in terms of
and
.
ANSWER:
=
This result is called the work-energy theorem. It states that the net work done on a particle equals the change in kinetic
energy of that particle.
Also notice that if
is zero, then the work-energy theorem reduces to
.
In other words, kinetic energy can be understood as the amount of work that is done to accelerate the particle from rest to

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