In Melde's experiment it was found that the string vibrated in three loops wien 8 gm were placed on the scale pan. What mass must be placed on the pan to make the string vibrate in six loops? (Neglect the mass of the string and the scale pan]

Answer

To solve the problem step by step, we can follow the reasoning laid out in the video transcript:

### Step 1: Understand the relationship between frequency, tension, and the number of loops
In Melde's experiment, the frequency of the vibrating string can be expressed as:
\[ f = \frac{N}{2L} \sqrt{\frac{T}{\mu}} \]
where:
- \( N \) is the number of loops,
- \( L \) is the length of the string,
- \( T \) is the tension in the string,
- \( \mu \) is the mass per unit length of the string.

### Step 2: Set up the equations for the two scenarios
1. For 3 loops (with mass \( m_1 = 8 \, \text{g} \)):
\[ f_1 = \frac{3}{2L} \sqrt{\frac{T_1}{\mu}} \]
The tension \( T_1 \) can be expressed as:
\[ T_1 = m_1 g \]
Therefore, substituting this into the frequency equation gives:
\[ f_1 = \frac{3}{2L} \sqrt{\frac{m_1 g}{\mu}} \]

2. For 6 loops (with mass \( m_2 \)):
\[ f_2 = \frac{6}{2L} \sqrt{\frac{T_2}{\mu}} \]
The tension \( T_2 \) can be expressed as:
\[ T_2 = m_2 g \]
Thus, substituting this into the frequency equation gives:
\[ f_2 = \frac{6}{2L} \sqrt{\frac{m_2 g}{\mu}} \]

### Step 3: Set the frequencies equal
Since the frequency must be the same for both scenarios (the string vibrates at the same frequency), we can set \( f_1 = f_2 \):
\[ \frac{3}{2L} \sqrt{\frac{m_1 g}{\mu}} = \frac{6}{2L} \sqrt{\frac{m_2 g}{\mu}} \]

### Step 4: Simplify the equation
We can cancel \( \frac{1}{2L} \) and \( g \) from both sides:
\[ 3 \sqrt{m_1} = 6 \sqrt{m_2} \]

### Step 5: Square both sides to eliminate the square root
Squaring both sides gives:
\[ 9 m_1 = 36 m_2 \]

### Step 6: Solve for \( m_2 \)
Rearranging the equation to solve for \( m_2 \):
\[ m_2 = \frac{9 m_1}{36} = \frac{m_1}{4} \]

### Step 7: Substitute the known value of \( m_1 \)
Substituting \( m_1 = 8 \, \text{g} \):
\[ m_2 = \frac{8}{4} = 2 \, \text{g} \]

### Final Answer
The mass that must be placed on the pan to make the string vibrate in six loops is:
\[ m_2 = 2 \, \text{g} \]

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