Question

A wheel of moment of inertia $I$ and radius $R$ is rotating about its axis at an angular speed ${\omega }_0$. It picks up a stationary particle of mass $m$ at its edge. Find the new angular speed of the wheel.

Answer 1

As the object is stationary and do not exert any force,the energy of the whole system remains conserved,i.e.
$(0.5\ast I\ast {\omega }^2{)}_{initial}=(0.5\ast I^{{}^{\prime }}\ast {\omega }^{{}^{\prime }2}{)}_{final}$
$I\ast {\omega }_0^2=(I+m{R}^2)\ast {\omega }^{{}^{\prime }2}$ from parallel axis theorem.
${\omega }^{{}^{\prime }}=\sqrt{\frac{I\ast {\omega }_0^2}{I+m{R}^2}}$
${\omega }^{{}^{\prime }}={\omega }_0\ast \sqrt{\frac{I}{I+m{R}^2}}$