In Melde's experiment , a string vibrates in 3 loops when 8 grams were placed in the pan.What mass must be placed in the pan to make the string vibrate in 5 loops ?

Answer

To solve the problem of determining the mass that must be placed in the pan to make the string vibrate in 5 loops, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between loops and tension**: In Melde's experiment, the number of loops (N) formed is related to the tension (T) in the string. The frequency of vibration is constant for a given setup.

2. **Use the formula for frequency**: The frequency of a vibrating string can be expressed as:
\[ f = \frac{N}{2L} \sqrt{\frac{T}{M}} \]
where:
- \( N \) = number of loops
- \( L \) = length of the string
- \( T \) = tension in the string (which is proportional to the mass hanging from the string)
- \( M \) = mass per unit length of the string

3. **Set up the ratio of frequencies for two different conditions**: Since the frequency is constant, we can set up the following equation for the two scenarios:
\[ \frac{N_1}{N_2} = \sqrt{\frac{T_2}{T_1}} \]
where:
- \( N_1 = 3 \) (for 3 loops with 8 grams)
- \( N_2 = 5 \) (for 5 loops)
- \( T_1 = 8 \) grams (initial tension)

4. **Plug in the values**: Substitute the known values into the equation:
\[ \frac{3}{5} = \sqrt{\frac{T_2}{8}} \]

5. **Square both sides to eliminate the square root**:
\[ \left(\frac{3}{5}\right)^2 = \frac{T_2}{8} \]
\[ \frac{9}{25} = \frac{T_2}{8} \]

6. **Solve for \( T_2 \)**: Multiply both sides by 8 to find \( T_2 \):
\[ T_2 = 8 \times \frac{9}{25} \]
\[ T_2 = \frac{72}{25} = 2.88 \text{ grams} \]

### Final Answer:
To make the string vibrate in 5 loops, a mass of **2.88 grams** must be placed in the pan.