Question

When two rigid bodies with moments of inertia \(I_1\) and \(I_2\) and angular velocities \(\omega_1\) and \(\omega_2\) respectively are coupled in such a way that their rotation axes coincide, the angular velocity of the combination is \(\omega\). Then

• A. \((I_1\omega_1+I_2\omega_2)=(I_1+I_2)\omega\)
• B. \(I_1\omega_1-I_2\omega_2=(I_1+I_2)\omega\)
• C. \(I_1I_2^2\omega_1\omega_2=I_1I_2\omega\)
• D. \((I_1+I_2)(\omega_1+\omega_2)=(I_1+I_2)\omega\)
• E. \(\omega_1+\omega_2=\omega\)

Hint:

In coupling of rotating bodies, always use conservation of angular momentum: initial total angular momentum = final total angular momentum.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This problem deals with the interaction of two rotating bodies. When they are coupled, they exert torques on each other. These are internal torques to the system of two bodies. If there is no external torque acting on the system, the total angular momentum of the system is conserved.
Step 2: Key Formula or Approach:
The principle of conservation of angular momentum states that if the net external torque on a system is zero, its total angular momentum remains constant.
\[ \vec{L}_{initial} = \vec{L}_{final} \] The angular momentum (\( \vec{L} \)) of a rigid body rotating about a fixed axis is given by \( \vec{L} = I\vec{\omega} \), where I is the moment of inertia and \( \vec{\omega} \) is the angular velocity.
Step 3: Detailed Explanation:
Initial State (Before Coupling):
The two bodies are rotating independently about the same axis.
- Angular momentum of the first body: \( L_1 = I_1 \omega_1 \)
- Angular momentum of the second body: \( L_2 = I_2 \omega_2 \)
Assuming they are rotating in the same direction, the total initial angular momentum of the system is the sum of their individual angular momenta:
\[ L_{initial} = L_1 + L_2 = I_1 \omega_1 + I_2 \omega_2 \] Final State (After Coupling):
The two bodies are coupled and rotate together as a single system.
- The total moment of inertia of the combined system is the sum of the individual moments of inertia: \( I_{final} = I_1 + I_2 \).
- The combined system rotates with a common final angular velocity, \( \omega \).
- The total final angular momentum of the system is: \( L_{final} = I_{final} \omega = (I_1 + I_2)\omega \).
Applying Conservation of Angular Momentum:
Since no external torque is mentioned, we assume the system is isolated. Therefore, the total angular momentum is conserved.
\[ L_{initial} = L_{final} \] \[ I_1 \omega_1 + I_2 \omega_2 = (I_1 + I_2)\omega \] This equation represents the conservation of angular momentum for the system.
Step 4: Final Answer:
The correct relation based on the conservation of angular momentum is \( I_1 \omega_1 + I_2 \omega_2 = (I_1 + I_2)\omega \). This corresponds to option (A).