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PHCC 141: Physics for Scientists and Engineers I - Fall 2007
6b. Work, Energy, and Power
Due at 11:59pm on Monday, October 1, 2007
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Integrate Force to Find Work
Work Done by a Spring
Consider a spring, with spring constant
, one end of which is attached to a wall. The spring is initially unstretched, with the
unconstrained end of the spring at position
.
Part A
The spring is now compressed so that the unconstrained end moves from
to
. Using the work integral
,
find the work done
by
the spring as it is compressed.
Hint A.1
Spring force as a function of position
The spring force vector
as a function of displacement
from the spring's equilibrium position, is given by
where
is the spring constant and
is a unit vector in the direction of the displacement of the spring (in this case, towards the
right).
Part A.2
Integrand of the work integral
The work done by the spring is given by the integral of the dot product of the spring force and an infinitesimal displacement of
the end of the spring:
,
where the infinitesmal displacement vector
has been written as
. Write
in terms of given quantities, and then compute
the dot product to find an expression for the integrand. (Note,
.)
Express your answer in terms of
,
, and
.
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Work Energy Problems
Work-Energy Scaling
A particle of mass
moves along a straight line with initial speed
. A force of magnitude
pushes the particle a distance
along the direction of its motion.
Part A
Find
, the final speed of the particle after it has traveled a distance
.
Part A.1
Find the final kinetic energy
Find
, the final kinetic energy of the particle after it has been pushed a distance
.
Part A.1.a
Find the final kinetic energy in terms of the work done
Assume that the work done by the pushing force is
. Find the final kinetic energy,
, of the particle after it has been pushed
a distance
.
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