First slide

Dynamics of rotational motion about fixed axis

Question

A ring of radius R is first rotated with an angular velocity ω₀ and then carefully placed on a rough horizontal surface. The coefficient of friction between the surface and the ring is µ. Time after which its angular speed is reduced to half is

Moderate

A

$\large \frac{{{\omega _0}\mu R}}{{2g}}$
B

$\large \frac{{{\omega _0}\mu R}}{{2\mu R}}$
C

$\large \frac{{{2\omega _0}R}}{{\mu g}}$
D

$\large \frac{{{\omega _0}R}}{{2\mu g}}$

Solution

I_C = MR², f_k = µMg

$\large \therefore \alpha =\frac {\tau}{I_C}=\frac {f_k.R}{MR^2}$

$\large =-\frac {\mu MgR}{MR^2}=-\frac {\mu g}{R}$

$\large \omega =\omega _0+\alpha t\Rightarrow \frac {\omega _0}{2}=\omega _0-\frac {\mu g}{R}t \Rightarrow t=\frac {\omega _0R}{2\mu g}$

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