AIPMT PRELIMS 2004 Physics Conservation of Angular Momentum MCQ Question

To solve this problem, we will apply the principle of conservation of angular momentum. The law states that if no external torque acts on a system, the total angular momentum of the system remains constant.

Given Information:

• Moment of inertia of the first disc (rotating disc): $I_1$
• Moment of inertia of the second disc (stationary disc): $I_2$
• Initial angular velocity of the first disc: $\omega$
• The second disc is placed over the first disc, and they rotate together after some time.

Initial Angular Momentum:

Before the second disc is placed on the first disc, the total angular momentum $L_{\text{initial}}$ of the system is solely due to the rotating disc:

$$L_{\text{initial}} = I_1 \cdot \omega$$

Final Angular Momentum:

After the second disc is placed on the first disc, both discs rotate together with a common final angular velocity $\omega_f$. The moment of inertia of the combined system is the sum of the individual moments of inertia:

$$I_{\text{total}} = I_1 + I_2$$

Thus, the final angular momentum $L_{\text{final}}$ of the system is given by:

$$L_{\text{final}} = I_{\text{total}} \cdot \omega_f = (I_1 + I_2) \cdot \omega_f$$

Applying Conservation of Angular Momentum:

According to the conservation of angular momentum, we can equate the initial and final angular momentum:

$$I_1 \cdot \omega = (I_1 + I_2) \cdot \omega_f$$

Solving for Final Angular Velocity:

Rearranging the equation to solve for $\omega_f$:

$$\omega_f = \frac{I_1 \cdot \omega}{I_1 + I_2}$$

This expression corresponds to option B:

$$\text{Final Angular Velocity} = \frac{I_1 \cdot \omega}{I_1 + I_2}$$

Explanation of Options:

• Option A: $\omega$

  • This implies that the final angular velocity remains the same as the initial angular velocity, which is incorrect because the second disc alters the moment of inertia of the system.

• Option B: $\frac{I_1 \cdot \omega}{I_1 + I_2}$

  • This is the correct answer as derived from the conservation of angular momentum.

• Option C: $\frac{(I_1 + I_2) \cdot \omega}{I_1}$

  • This suggests that the final angular velocity would increase based on the total moment of inertia, which is incorrect as it does not account for the fact that the second disc is initially stationary.

• Option D: $\frac{I_2 \cdot \omega}{I_1 + I_2}$

  • This option incorrectly relates the final angular velocity to the moment of inertia of the stationary disc, not accounting for the initial motion of the first disc.

Conclusion:

Hence, the correct answer is option B, $\frac{I_1 \cdot \omega}{I_1 + I_2}$, which accurately reflects the change in angular velocity due to the increase in the moment of inertia when the second disc is added. This example effectively demonstrates the application of the conservation of angular momentum in rotational dynamics.