Question

a solid sphere of radius 'r' is revolving about one of its diameter with an angular velocity 'w'.If it suddenly expands uniformly so that it's radius increases to n times its original value, then it's angular velocity becomes __​

Answer 1

Given:

a solid sphere of radius 'r' is revolving about one of its diameter with an angular velocity 'w'.If it suddenly expands uniformly so that it's radius increases to n times its original value.

To find:

Value of new angular velocity?

Calculation:

• Since there is no external torque acting on the system, the angular momentum of the system will remain constant

$L_{1} = L_{2}$

$\implies \: I_{1} \omega_{1} = I_{2} \omega_{2}$

$\implies \: (\dfrac{2}{5}m {r}^{2}) \omega_{1} = \{ \dfrac{2}{5} m {(nr)}^{2} \} \omega_{2}$

$\implies \: {r}^{2} \omega_{1} = {n}^{2} {r}^{2} \omega_{2}$

$\implies \: \omega_{1} = {n}^{2} \omega_{2}$

$\implies \: \omega_{2} = \dfrac{ \omega_{1} }{ {n}^{2} }$

$\implies \: \omega_{2} = \dfrac{ \omega }{ {n}^{2} }$

So, final angular velocity is :

$\boxed{ \bf\: \omega_{2} = \dfrac{ \omega }{ {n}^{2} } }$