Question

A disc of moment of inertia 4 kg \( \text{m}^2 \) revolving with 16 rad/s is placed on another disc of moment of inertia 8 kg \( \text{m}^2 \) revolving with 4 rad/s. The angular frequency of the composite disc is

• A. 4 rad/s
• B. \( \frac{3}{16} \) rad/s
• C. 8 rad/s
• D. \( \frac{16}{3} \) rad/s

Hint:

When combining rotating objects, use the conservation of angular momentum to find the final angular velocity. The total angular momentum before the combination equals the total angular momentum after the combination.

Answer 1

The Correct Option is C

The system described involves two discs with different moments of inertia and angular velocities. To find the angular frequency of the composite disc, we apply the principle of conservation of angular momentum. The total angular momentum of the system before the discs are combined is the sum of the individual angular momenta of the discs. The angular momentum \( L \) of an object is given by: \[ L = I \cdot \omega \] where \( I \) is the moment of inertia and \( \omega \) is the angular velocity. - For the first disc: \[ L_1 = I_1 \cdot \omega_1 = 4 \cdot 16 = 64 \, \text{kg} \cdot \text{m}^2/\text{s} \] - For the second disc: \[ L_2 = I_2 \cdot \omega_2 = 8 \cdot 4 = 32 \, \text{kg} \cdot \text{m}^2/\text{s} \] The total angular momentum of the system is: \[ L_{\text{total}} = L_1 + L_2 = 64 + 32 = 96 \, \text{kg} \cdot \text{m}^2/\text{s} \] Since angular momentum is conserved, the total angular momentum after the discs are combined is equal to the total angular momentum before. The combined moment of inertia is: \[ I_{\text{total}} = I_1 + I_2 = 4 + 8 = 12 \, \text{kg} \cdot \text{m}^2 \] Thus, the final angular velocity \( \omega_{\text{final}} \) is: \[ \omega_{\text{final}} = \frac{L_{\text{total}}}{I_{\text{total}}} = \frac{96}{12} = 8 \, \text{rad/s} \] Therefore, the angular frequency of the composite disc is 8 rad/s, so the correct answer is (C).