A string 1 m long is fixed at one end. The other end is moved up and down with frequency 20 Hz. Due to this, a stationary wave with four complete loops, gets produced on the string. Find the speed of the progressive wave which produces the stationary wave. [Note: Remember that the moving end is a antinode.]

Answer

To solve the problem, we need to find the speed of the progressive wave that produces the stationary wave on a string that is fixed at one end and has a moving end that acts as an antinode.

### Step-by-Step Solution:

1. **Understand the Setup**:
- The string is 1 meter long and fixed at one end.
- The other end is moved up and down with a frequency of 20 Hz.
- The stationary wave produced has 4 complete loops (or wavelengths).

2. **Identify the Antinode and Node**:
- Since the fixed end is a node and the moving end is an antinode, the stationary wave will have a pattern where the antinode is at the moving end.

3. **Determine the Length of the String in Terms of Wavelength**:
- For a stationary wave with \( n \) loops, the relationship between the length \( L \) of the string and the wavelength \( \lambda \) is given by:
\[ L = \frac{n \lambda}{2} \]
- Here, \( n = 4 \) (for 4 complete loops).
- Therefore, the equation becomes:
\[ L = \frac{4 \lambda}{2} = 2\lambda \]

4. **Account for the Antinode**:
- Since the moving end is an antinode, we need to add an additional half wavelength to the length of the string:
\[ L = 2\lambda + \frac{\lambda}{2} = 2.5\lambda = \frac{5\lambda}{2} \]

5. **Solve for Wavelength**:
- Given that the length of the string \( L = 1 \) meter, we can set up the equation:
\[ 1 = \frac{5\lambda}{2} \]
- Rearranging gives:
\[ \lambda = \frac{2}{5} \text{ meters} = 0.4 \text{ meters} \]

6. **Calculate the Speed of the Wave**:
- The speed \( v \) of a wave is given by the formula:
\[ v = f \lambda \]
- Where \( f \) is the frequency. Given \( f = 20 \) Hz:
\[ v = 20 \times 0.4 = 8 \text{ m/s} \]

### Final Answer:
The speed of the progressive wave that produces the stationary wave is **8 m/s**.