Two springs of constants `k_1` and `k_2` have equal maximum velocities, when executing simple harmonic motion. The ratio of their amplitudes (masses are equal) will be

Options

A. `(k_1/k_2)^(1/2)`
B. `(k_1/k_2)`
C. `(k_2/k_1)`
D. `(k_2/k_1)^(1/2)`

Answer

To solve the problem, we need to analyze the relationship between the maximum velocities of two springs executing simple harmonic motion (SHM) and their respective amplitudes. Given that the masses are equal, we can derive the ratio of their amplitudes based on their spring constants.

### Step-by-Step Solution:

1. **Understanding the Kinetic Energy in SHM**:
The kinetic energy (KE) of an object executing SHM is given by the formula:
\[ KE = \frac{1}{2} m v_{\text{max}}^2 \]
where \( m \) is the mass and \( v_{\text{max}} \) is the maximum velocity.

2. **Understanding the Potential Energy in SHM**:
The potential energy (PE) stored in a spring when it is stretched or compressed is given by:
\[ PE = \frac{1}{2} k a^2 \]
where \( k \) is the spring constant and \( a \) is the amplitude of the motion.

3. **Setting Up the Equations**:
For the two springs with constants \( k_1 \) and \( k_2 \), and amplitudes \( a_1 \) and \( a_2 \), we can write the kinetic energy equations as follows:
- For spring 1:
\[ \frac{1}{2} m v_{\text{max}}^2 = \frac{1}{2} k_1 a_1^2 \]
- For spring 2:
\[ \frac{1}{2} m v_{\text{max}}^2 = \frac{1}{2} k_2 a_2^2 \]

4. **Equating the Kinetic Energies**:
Since the maximum velocities are equal for both springs, we can set the two equations equal to each other:
\[ \frac{1}{2} k_1 a_1^2 = \frac{1}{2} k_2 a_2^2 \]

5. **Simplifying the Equation**:
Canceling the \(\frac{1}{2}\) from both sides gives:
\[ k_1 a_1^2 = k_2 a_2^2 \]

6. **Finding the Ratio of Amplitudes**:
Rearranging the equation to find the ratio of the amplitudes \( \frac{a_1}{a_2} \):
\[ \frac{a_1^2}{a_2^2} = \frac{k_2}{k_1} \]
Taking the square root of both sides, we get:
\[ \frac{a_1}{a_2} = \sqrt{\frac{k_2}{k_1}} \]

7. **Final Answer**:
Thus, the ratio of their amplitudes is:
\[ \frac{a_1}{a_2} = \sqrt{\frac{k_2}{k_1}} \]

### Conclusion:
The final answer is \( \frac{a_1}{a_2} = \sqrt{\frac{k_2}{k_1}} \).