Question

Two discs of same moment of inertia rotating about their regular axis passing through centre and perpendicular to the plane of disc with angular velocities $\omega_1$ and $\omega_2$ . They are brought into contact face to face coinciding the axis of rotation. The expression for loss of energy during this process is:-

• A. $\frac{1}{4} I (\omega_1 - \omega_2)^2$
• B. $I (\omega_1 - \omega_2)^2$
• C. $\frac{1}{8} I (\omega_1 - \omega_2)^2$
• D. $\frac{1}{2} I (\omega_1+ \omega_2)^2$

Answer 1

The Correct Option is A

To find the loss of energy when two discs with the same moment of inertia are brought into contact, we start by understanding the concept of conservation of angular momentum and kinetic energy in rotational systems.

1. The moment of inertia of both discs is the same, say I.
2. Initially, the first disc is rotating with angular velocity \omega_1 and the second disc with \omega_2.
3. When two discs are brought into contact, due to friction, they will eventually rotate with a common angular velocity \omega_f.
4. By conservation of angular momentum:
   I \omega_1 + I \omega_2 = 2I \omega_f
5. Solving for \omega_f:
   \omega_f = \frac{\omega_1 + \omega_2}{2}
6. The initial kinetic energy of the system is:
   K.E._{initial} = \frac{1}{2} I \omega_1^2 + \frac{1}{2} I \omega_2^2
7. The final kinetic energy when both discs rotate together:
   K.E._{final} = \frac{1}{2} (2I) \omega_f^2 = I \left(\frac{\omega_1 + \omega_2}{2}\right)^2
8. Simplifying:
   K.E._{final} = \frac{I}{2} \left(\omega_1^2 + 2\omega_1\omega_2 + \omega_2^2 \right)/2 = \frac{I}{4} (\omega_1^2 + 2\omega_1\omega_2 + \omega_2^2)
9. The loss of energy during the process = K.E._{initial} - K.E._{final}:
   \begin{align*} \Delta E &= \left(\frac{1}{2} I \omega_1^2 + \frac{1}{2} I \omega_2^2\right) - \frac{I}{4} (\omega_1^2 + 2\omega_1\omega_2 + \omega_2^2)\\ &= \frac{1}{4} I (\omega_1^2 + \omega_2^2 - 2\omega_1\omega_2)\\ &= \frac{1}{4} I (\omega_1 - \omega_2)^2 \end{align*}
10. Therefore, the expression for the loss of energy during this process is:
    \frac{1}{4} I (\omega_1 - \omega_2)^2

Thus, the correct option is: \frac{1}{4} I (\omega_1 - \omega_2)^2.