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Rotational Inertia

A uniform rin...

Question

A uniform ring of radius R is given a backspin of angular velocity $V_{o} / 2 R$ and thrown on a horizontal rough surface with the velocity of the centre to be $V_{o}$ . Find the velocity of the centre of the ring when it starts pure rolling.

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Solution

We know about point any point $P$ on ground Angular momentum will be conserved. As torque due to friction blows off.
$V = r w$
$I_{c m} = M v^{2} v_{i} = v_{0} v_{f} = v w_{i} = - \frac{v_{0}}{2 R} w_{f} = w_{p} = c o n s t a n t m v_{0} r - m r^{2} \times \frac{v_{0}}{2 v} = m v r + m r^{2} w \Rightarrow v_{0} - \frac{v_{0}}{2} = 2 v \Rightarrow v = \frac{v_{0}}{4}$
$∴$ Final velocity is $\frac{v_{0}}{4}$ in forward direction.

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