Two bodies (M) and (N) of equal masses are suspended from two separate massless springs of spring constants (k_1) and (k_2) respectively. If the two bodies oscillate vertically such that their maximum velocities are equal, the ratio of the amplitude of vibration of (M) to the of (N) is.

Options

A. `(k_(1))/(k_(2))`
B. `sqrt((k_(1))/(k_(2)))`
C. `(k_(2))/(k_(1))`
D. `sqrt((k_(2))/(k_(1)))`

Explanation

To solve the problem step by step, we will analyze the given information and apply the relevant formulas for simple harmonic motion (SHM).

### Step 1: Understand the System
We have two bodies, M and N, each suspended from separate massless springs with spring constants \( k_1 \) and \( k_2 \) respectively. Both bodies have equal mass \( m \).

### Step 2: Identify the Maximum Velocity in SHM
The maximum velocity \( v_{max} \) of a body in simple harmonic motion is given by the formula:
\[ v_{max} = A \omega \]
where \( A \) is the amplitude of the oscillation and \( \omega \) is the angular frequency.

### Step 3: Express Angular Frequency
The angular frequency \( \omega \) for a mass-spring system is given by:
\[ \omega = \sqrt{\frac{k}{m}} \]
Thus, for body M:
\[ \omega_1 = \sqrt{\frac{k_1}{m}} \]
And for body N:
\[ \omega_2 = \sqrt{\frac{k_2}{m}} \]

### Step 4: Set Up the Equation for Maximum Velocities
Since it is given that the maximum velocities of both bodies are equal, we can write:
\[ A_1 \omega_1 = A_2 \omega_2 \]
Substituting the expressions for \( \omega_1 \) and \( \omega_2 \):
\[ A_1 \sqrt{\frac{k_1}{m}} = A_2 \sqrt{\frac{k_2}{m}} \]

### Step 5: Simplify the Equation
We can cancel \( m \) from both sides:
\[ A_1 \sqrt{k_1} = A_2 \sqrt{k_2} \]

### Step 6: Solve for the Ratio of Amplitudes
Rearranging the equation gives:
\[ \frac{A_1}{A_2} = \frac{\sqrt{k_2}}{\sqrt{k_1}} \]
Thus, the ratio of the amplitudes of vibration of M to that of N is:
\[ \frac{A_1}{A_2} = \sqrt{\frac{k_2}{k_1}} \]

### Final Answer
The ratio of the amplitude of vibration of M to that of N is:
\[ \frac{A_1}{A_2} = \sqrt{\frac{k_2}{k_1}} \]