First slide

Rotational motion

Question

In the following figure, a body of mass m is tied at one end of a light string and this string is wrapped around the solid cylinder of mass M and radius R. At the moment t=0,the system starts moving. If the friction is negligible, angular velocity at time t would be

Moderate

A

$\frac{m g R t}{(M + m)}$
B

$\frac{2 M g t}{(M + 2 m)}$
C

$\frac{2 m g t}{R (M - 2 m)}$
D

$\frac{2 m g t}{R (M + 2 m)}$

Solution

We know the tangential acceleration
$a = \frac{g}{1 + \frac{I}{m R^{2}}} = \frac{g}{1 + \frac{1 / 2 M R^{2}}{m R^{2}}} = \frac{2 m g}{2 m + M}$
$[As I = \frac{1}{2} M R^{2} for cylinder]$
After time t, linear velocity of mass m, $v = u + a t = 0 + \frac{2 m g t}{2 m + M}$
So, angular velocity of the cylinder, $ω = \frac{v}{R} = \frac{2 m g t}{R (M + 2 m)}$

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