Question

A disc of moment of inertia \(I_1\) is rotating with angular velocity \(\omega_1\) about an axis perpendicular to its plane passing through its centre. If another disc of moment of inertia \(I_2\) about the same axis is gently placed over it, then the new angular velocity of the combined disc will be

• A. \( \dfrac{I_1 \omega_1}{I_1 + I_2} \)
• B. \( \dfrac{(I_1 + I_2)\omega_1}{I_1} \)
• C. \( \dfrac{I_2 \omega_1}{I_1 + I_2} \)
• D. \( \omega_1 \)

Hint:

In problems where objects stick together while rotating, always apply conservation of angular momentum.

Answer 1

The Correct Option is A

Step 1: Principle used.
Since no external torque acts on the system, angular momentum is conserved.
Step 2: Initial angular momentum.
Initially, only the first disc is rotating. Hence,
\[ L_i = I_1 \omega_1 \]
Step 3: Final angular momentum.
After the second disc is gently placed, both discs rotate together with angular velocity \(\omega\).
\[ L_f = (I_1 + I_2)\omega \]
Step 4: Applying conservation of angular momentum.
\[ I_1 \omega_1 = (I_1 + I_2)\omega \]
Step 5: Solving for \(\omega\).
\[ \omega = \frac{I_1 \omega_1}{I_1 + I_2} \]
Step 6: Conclusion.
The new angular velocity of the combined disc is \( \dfrac{I_1 \omega_1}{I_1 + I_2} \).