Question

The angular velocities of three bodies in simple harmonic motion are \(\omega_{1},\omega_{2},\omega_{3}\) with their respective amplitudes as \(A_{1},A_{2},A_{3}\). If all the three bodies have same mass and velocity, then

• A. \(A_1^2\omega_1^2 = A_2^2\omega_2^2 = A_3^2\omega_3^2\)
• B. \(A_1^2\omega_1 = A_2^2\omega_2 = A_3^2\omega_3\)
• C. \(A_1\omega_1^2 = A_2\omega_2^2 = A_3\omega_3^2\)
• D. \(A_1\omega_1 = A_2\omega_2 = A_3\omega_3\)

Hint:

In SHM, maximum velocity \(v_{\text{max}} = A\omega\). When velocities are equal, amplitudes and angular frequencies are inversely related.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
For SHM, displacement \(y = A\sin\omega t\). Velocity \(v = \frac{dy}{dt} = A\omega\cos\omega t\). Maximum velocity \(v_{\text{max}} = A\omega\).
Step 2: Detailed Explanation:
Given masses are equal and velocities are equal. In SHM, when velocity is maximum, \(v_{\text{max}} = A\omega\). Since velocity is same for all three bodies, \(A_1\omega_1 = A_2\omega_2 = A_3\omega_3\).
Step 3: Final Answer:
Thus, \(A_1\omega_1 = A_2\omega_2 = A_3\omega_3\).