We give a strategy for using this equation when analyzing rotational motion.
Problem-solving strategy: work-energy theorem for rotational motion
Identify the forces on the body and draw a free-body diagram. Calculate the torque for each force.
Calculate the work done during the body’s rotation by every torque.
Apply the work-energy theorem by equating the net work done on the body to the change in rotational kinetic energy.
Let’s look at two examples and use the work-energy theorem to analyze rotational motion.
Rotational work and energy
A
torque is applied to a flywheel that rotates about a fixed axis and has a moment of inertia of
. If the flywheel is initially at rest, what is its angular velocity after it has turned through eight revolutions?
Strategy
We apply the work-energy theorem. We know from the problem description what the torque is and the angular displacement of the flywheel. Then we can solve for the final angular velocity.
Solution
The flywheel turns through eight revolutions, which is
radians. The work done by the torque, which is constant and therefore can come outside the integral in
[link] , is
We apply the work-energy theorem:
With
, we have
Therefore,
This is the angular velocity of the flywheel after eight revolutions.
Significance
The work-energy theorem provides an efficient way to analyze rotational motion, connecting torque with rotational kinetic energy.
A string wrapped around the pulley in
[link] is pulled with a constant downward force
of magnitude 50 N. The radius
R and moment of inertia
I of the pulley are 0.10 m and
, respectively. If the string does not slip, what is the angular velocity of the pulley after 1.0 m of string has unwound? Assume the pulley starts from rest.
Strategy
Looking at the free-body diagram, we see that neither
, the force on the bearings of the pulley, nor
, the weight of the pulley, exerts a torque around the rotational axis, and therefore does no work on the pulley. As the pulley rotates through an angle
acts through a distance
d such that
Solution
Since the torque due to
has magnitude
, we have
If the force on the string acts through a distance of 1.0 m, we have, from the work-energy theorem,
Power always comes up in the discussion of applications in engineering and physics. Power for rotational motion is equally as important as power in linear motion and can be derived in a similar way as in linear motion when the force is a constant. The linear power when the force is a constant is
. If the net torque is constant over the angular displacement,
[link] simplifies and the net torque can be taken out of the integral. In the following discussion, we assume the net torque is constant. We can apply the definition of power derived in
Power to rotational motion. From
Work and Kinetic Energy , the instantaneous power (or just power) is defined as the rate of doing work,
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