First slide

Dynamics of rotational motion about fixed axis

Question

A string is wrapped around a wheel of radius ‘r’. The axis of the wheel is horizontal and its M.I. about the axis is I. Weight ‘mg’ is tied to free end of the string which is released to fall down from rest position. The angular velocity of the wheel after it has fallen through distance ‘h’ will be

Moderate

A

$\large {\left( {\frac{{2gh}}{{I + m{r^2}}}} \right)^{\frac{1}{2}}}$
B

$\large {\left( {\frac{{2mgh}}{{I + m{r^2}}}} \right)^{\frac{1}{2}}}$
C

$\large \frac{{2mgh}}{{I + 2m{r^2}}}$
D

$\large \sqrt {2gh}$

Solution

Loss in P.E. of the system = Gain in KE of the system.
$\large mgh = \frac{1}{2}m{v^2} + \frac{1}{2}I{\omega ^2} = \frac{1}{2}\left[ {m{r^2} + I} \right]{\omega ^2}\$
$\large \omega = {\left( {\frac{{2mgh}}{{m{r^2} + I}}} \right)^{1/2}}\left[ {\because V = r\omega } \right]$

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