Two bodies `P` and `Q` of equal masses are suspended from two separate massless springs of force 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 `P` to that of `Q` is

Options

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

Explanation

To solve the problem step by step, we need to analyze the given conditions and apply the concepts of simple harmonic motion (SHM).

### Step 1: Understanding the given information
We have two bodies, P and Q, each with equal masses (let's denote the mass as \( m \)). They are suspended from springs with spring constants \( k_1 \) and \( k_2 \), respectively. Both bodies oscillate vertically and have the same maximum velocities.

### Step 2: Relating maximum velocity to amplitude
In SHM, the maximum velocity (\( V_{\text{max}} \)) of a mass-spring system is given by the formula:
\[ V_{\text{max}} = \omega A \]
where:
- \( \omega \) is the angular frequency,
- \( A \) is the amplitude of oscillation.

### Step 3: Expressing angular frequency
The angular frequency \( \omega \) for a mass-spring system is given by:
\[ \omega = \sqrt{\frac{k}{m}} \]
where \( k \) is the spring constant and \( m \) is the mass.

### Step 4: Setting up the equations
For body P:
\[ V_{1, \text{max}} = \omega_1 A_1 = \sqrt{\frac{k_1}{m}} A_1 \]
For body Q:
\[ V_{2, \text{max}} = \omega_2 A_2 = \sqrt{\frac{k_2}{m}} A_2 \]
Since \( V_{1, \text{max}} = V_{2, \text{max}} \), we can equate the two expressions:
\[ \sqrt{\frac{k_1}{m}} A_1 = \sqrt{\frac{k_2}{m}} A_2 \]

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

### Step 6: Finding the ratio of amplitudes
Rearranging the equation gives us:
\[ \frac{A_1}{A_2} = \frac{\sqrt{k_2}}{\sqrt{k_1}} \]
Thus, the ratio of the amplitudes of P to Q is:
\[ \frac{A_1}{A_2} = \sqrt{\frac{k_2}{k_1}} \]

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

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