The vibrations from an 800 Hz tuning fork set up standing waves in a string clamped at both ends. The wave speed in the string is known to be `400 m//s ` for the tension used. The standing wave is observed to have four antinodes. How long is the string?

Explanation

To find the length of the string that supports standing waves with four antinodes, we can use the relationship between frequency, wave speed, and the length of the string in the context of standing waves.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Frequency \( f = 800 \, \text{Hz} \)
- Wave speed \( v = 400 \, \text{m/s} \)
- Number of antinodes \( n = 4 \) (which corresponds to the 4th harmonic)

2. **Use the Formula for Frequency in Terms of Harmonics:**
The frequency of a string clamped at both ends can be expressed as:
\[ f = \frac{n v}{2l} \]
where:
- \( n \) is the harmonic number (4 in this case),
- \( v \) is the wave speed,
- \( l \) is the length of the string.

3. **Rearrange the Formula to Solve for Length \( l \):**
Rearranging the formula gives:
\[ l = \frac{n v}{2f} \]

4. **Substitute the Known Values:**
Now, substitute the known values into the equation:
\[ l = \frac{4 \times 400}{2 \times 800} \]

5. **Calculate the Length:**
- First, calculate the numerator:
\[ 4 \times 400 = 1600 \]
- Then, calculate the denominator:
\[ 2 \times 800 = 1600 \]
- Now, divide the numerator by the denominator:
\[ l = \frac{1600}{1600} = 1 \, \text{m} \]

6. **Final Answer:**
The length of the string is \( l = 1 \, \text{m} \).