Question

Two discs with the same mass and radii of l and 2l are mounted co-axially on the same vertical axis. Disc A is impareted an initial angular velocity of 2ω and disc B is imparted an initial velocity of ω. Both discs are rotating in clockwise direction. If disc B is brought in contact with disc A, then what is the common angular velocity acquired by them?
1) ω/3
2) 4ω/3
3) 2ω/3
4) 5ω/3

Answer 1

Moment of inertia of disc A, $\sf{I_1=I.}$

Moment of inertia of disc B, $\sf{I_2=2I.}$

Initial angular velocity of disc A, $\sf{\omega_{1}=2\omega.}$

Initial angular velocity of disc B, $\sf{\omega_{2}=\omega.}$

Both disc are rotating in clockwise direction.

Let the final angular velocity of the system after bringing disc B in contact with disc A be $\sf{\omega'.}$

As the discs are mounted co-axially on the same vertical axis, the net moment of inertia of the system will be $\sf{I_1+I_2.}$

Then by conservation of angular momentum (net torque acting on the system is zero),

$\sf{\longrightarrow I_1\omega_{1}+I_2\omega_2=(I_1+I_2)\,\omega'}$

$\sf{\longrightarrow\omega'=\dfrac{I_1\omega_{1}+I_2\omega_2}{I_1+I_2}}$

$\sf{\longrightarrow\omega'=\dfrac{2I\omega+2I\omega}{I+2I}}$

$\sf{\longrightarrow\underline{\underline{\omega'=\dfrac{4\omega}{3}}}}$

Hence (2) is the answer.