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: Understand the physical situation.
Two rigid bodies are rotating about the same axis with angular velocities \(\omega_1\) and \(\omega_2\), and their moments of inertia are \(I_1\) and \(I_2\). After they are coupled together, they rotate as one combined system with common angular velocity \(\omega\).

Step 2: Identify the physical principle involved.
When two rotating bodies are coupled and no external torque acts on the system, the total angular momentum of the system remains conserved. This is the law of conservation of angular momentum.

Step 3: Write the initial angular momentum of the system.
Before coupling, the first body has angular momentum:
\[ L_1=I_1\omega_1 \] and the second body has angular momentum:
\[ L_2=I_2\omega_2 \] Therefore, total initial angular momentum is:
\[ L_{\text{initial}}=I_1\omega_1+I_2\omega_2 \]

Step 4: Write the final angular momentum of the combined system.
After coupling, both bodies rotate together with common angular velocity \(\omega\). The total moment of inertia of the combined system is:
\[ I_1+I_2 \] Hence, the final angular momentum is:
\[ L_{\text{final}}=(I_1+I_2)\omega \]

Step 5: Apply conservation of angular momentum.
Since no external torque acts, we must have:
\[ L_{\text{initial}}=L_{\text{final}} \] So, \[ I_1\omega_1+I_2\omega_2=(I_1+I_2)\omega \]

Step 6: Compare with the given options.
The obtained relation exactly matches option \((1)\). This is the standard expression for the common angular velocity after coupling of two rotating bodies about the same axis.

Step 7: State the final answer.
Thus, the correct relation is:
\[ \boxed{I_1\omega_1+I_2\omega_2=(I_1+I_2)\omega} \] which matches option \((1)\).