Question

A disc of the momentum of inertia | rotating about its own axis with angular speed m. It is placed on another disc of the moment of inertia 31 which is at rest. Both disc have a common axis of rotation. What will be the loss of kinetic energy up to both discs attain common angular velocity?

Answer 1

It has given that, moment of inertia of first disc = I, about its own axis. angular speed of first disc = ω.

moment of inertia of 2nd disc = 3I , this disc is at rest.

To find : if both discs have common axis of rotation, the loss in kinetic energy upto both discs attain common angular velocity.

solution : as external torque equals to zero.

from conservation of angular momentum,

$I_1\omega_1+I_2\omega_2=I_c\omega_c$

here, $I_1=I,I_2=3I,\omega_1=\omega,\omega_2=0$

$I_c=I+3I=4I$

now, Iω + 3I × 0 = (I + 3I)$\omega_c$

$\omega_c$ = ω/4

now loss in kinetic energy = $\frac{1}{2}I_1\omega_1^2+\frac{1}{2}I_2\omega_2^2-\frac{1}{2}I_c^2\omega_c^2$

= 1/2 × I × ω² + 0 - 1/2 × 4I × (ω/4)²

= 1/2 Iω² [ 1 - 1/4]

= (3/8)Iω²

Therefore the loss in kinetic energy is (3/8)Iω² i.e., correct option is (3)