Kevin S. Huang

2007: Problem 11

Topic: Rigid Bodies
Concepts: Angular kinematics, Kinetic energy, Moments of inertia

Solution:

Recall for rotational motion,

\tau=I\alpha

From kinematics,

\omega=\omega_0+\alpha t=\frac{\tau t}{I}

Thus, the kinetic energy of the objects after time $t$ is given by

K=\frac{1}{2}I\omega^2=\frac{1}{2}I\left(\frac{\tau t}{I}\right)^2=\frac{\tau^2t^2}{2I}

The torques on all the objects are the same so

K \propto \frac{1}{I}

Recall the moments of inertia:

I_{\text{disk}}=\frac{1}{2}MR^2

I_{\text{hoop}}=MR^2

I_{\text{sphere}}=\frac{2}{5}MR^2

Hence,

K_{\text{hoop}} < K_{\text{disk}} < K_{\text{sphere}}

so the answer is E.