A block suspended from a vertical spring is in equilibrium. Show that the extension of the spring equals the length of an equivalent simple pendulum, i.e., a pendulum having frequency same as that of the block.

Explanation

To solve the problem of showing that the extension of a spring (x₀) equals the length of an equivalent simple pendulum (L) with the same frequency, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Forces**:
- A block of mass \( m \) is suspended from a vertical spring. At equilibrium, the weight of the block (mg) is balanced by the restoring force of the spring (kx₀), where k is the spring constant and x₀ is the extension of the spring.
- Therefore, at equilibrium, we have:
\[ kx₀ = mg \]

2. **Finding the Extension**:
- Rearranging the equation from Step 1 gives us the extension of the spring:
\[ x₀ = \frac{mg}{k} \]

3. **Frequency of the Spring-Mass System**:
- The frequency \( f \) of the spring-mass system is given by:
\[ f = \frac{1}{2\pi} \sqrt{\frac{k}{m}} \]

4. **Frequency of the Simple Pendulum**:
- The frequency of a simple pendulum is given by:
\[ f = \frac{1}{2\pi} \sqrt{\frac{g}{L}} \]
- Where \( g \) is the acceleration due to gravity and \( L \) is the length of the pendulum.

5. **Equating Frequencies**:
- Since the frequencies of both systems are equal, we can set their expressions equal to each other:
\[ \frac{1}{2\pi} \sqrt{\frac{k}{m}} = \frac{1}{2\pi} \sqrt{\frac{g}{L}} \]

6. **Simplifying the Equation**:
- Canceling \( \frac{1}{2\pi} \) from both sides, we get:
\[ \sqrt{\frac{k}{m}} = \sqrt{\frac{g}{L}} \]

7. **Squaring Both Sides**:
- Squaring both sides gives:
\[ \frac{k}{m} = \frac{g}{L} \]

8. **Rearranging to Find L**:
- Rearranging the equation to solve for L, we have:
\[ L = \frac{mg}{k} \]

9. **Substituting for x₀**:
- From Step 2, we know that \( x₀ = \frac{mg}{k} \). Therefore, we can substitute this into our equation for L:
\[ L = x₀ \]

### Conclusion:
We have shown that the extension of the spring \( x₀ \) equals the length of an equivalent simple pendulum \( L \).