Question

A wheel rolling along a rough horizontal surface with an angular velocity $\omega$ strikes a rough vertical wall, normally. Find the initial angular velocity, as it tends to roll up the wall (neglect any slippage) due to impulse.

Answer 1

R.E.F image

impulse $=\Delta p$

$=mV-O$ normal to vertical wall

impulsive friction $=\mu mv$

moment of impulsive friction $=\left(\mu mv\right)R=I\left(\Delta \omega \right)$

$\mu \frac{m\omega R^2}{I}={\omega }_f-\omega$

${\omega }_f=\omega (1+\frac{\mu m{R}^2}{I})$

wheel $I=m{R}^2$

${\omega }_f=\omega (1+\mu )$

If $I=\frac{m{R}^2}{2}$

${\omega }_f=\omega (1+\frac{m}{2})$