Two bodies A and B of equal mass are suspended from two separate massless springs of spring constant `k_1 and k_2` respectively. If the bodies Oscillate vertically such that their maximum velocities are equal, the ratio of the amplitude of A to that of B is

Options

A. `k_1/k_2`
B. `sqrt(k_1/k_2)`
C. `k_2/k_1`
D. `sqrt(k_2/k_1)`

Answer

To solve the problem, we need to find the ratio of the amplitudes of two bodies A and B that are oscillating vertically on springs with different spring constants, given that their maximum velocities are equal.

### Step-by-Step Solution:

1. **Understanding Maximum Velocity in SHM**:
The maximum velocity \( V_{max} \) of a body in simple harmonic motion (SHM) is given by the formula:
\[ V_{max} = \omega A \]
where \( \omega \) is the angular frequency and \( A \) is the amplitude.

2. **Setting Up the Problem**:
Let:
- Amplitude of body A = \( A \)
- Amplitude of body B = \( B \)
- Spring constant of body A = \( k_1 \)
- Spring constant of body B = \( k_2 \)
- Mass of both bodies = \( m \) (equal for both)

3. **Equating Maximum Velocities**:
Since the maximum velocities of both bodies are equal, we can write:
\[ \omega_A A = \omega_B B \]

4. **Finding Angular Frequencies**:
The angular frequency \( \omega \) for a mass-spring system is given by:
\[ \omega = \sqrt{\frac{k}{m}} \]
Therefore, we can express \( \omega_A \) and \( \omega_B \) as:
\[ \omega_A = \sqrt{\frac{k_1}{m}} \quad \text{and} \quad \omega_B = \sqrt{\frac{k_2}{m}} \]

5. **Substituting Angular Frequencies**:
Substitute \( \omega_A \) and \( \omega_B \) into the equation for maximum velocities:
\[ \sqrt{\frac{k_1}{m}} A = \sqrt{\frac{k_2}{m}} B \]

6. **Simplifying the Equation**:
We can cancel \( \sqrt{m} \) from both sides:
\[ \sqrt{k_1} A = \sqrt{k_2} B \]

7. **Finding the Ratio of Amplitudes**:
Rearranging gives us:
\[ \frac{A}{B} = \frac{\sqrt{k_2}}{\sqrt{k_1}} \]
Thus, the ratio of the amplitudes is:
\[ \frac{A}{B} = \sqrt{\frac{k_2}{k_1}} \]

### Final Answer:
The ratio of the amplitude of A to that of B is:
\[ \frac{A}{B} = \sqrt{\frac{k_2}{k_1}} \]