Two strings of the same material and the same area of cross-section are used in sonometer experiment. One is loaded with 12kg and the other with 3 kg. The fundamental frequency of the first string is equal to the first overtone of the second string. If the length of the second string is 100 cm, then the length of the first string is

Options

A. 300cm
B. 200cm
C. 100cm
D. 50cm

Answer

To find the length of the first string in the sonometer experiment, we can follow these steps:

### Step 1: Understand the relationship between frequency and tension
The fundamental frequency \( f \) of a string is given by the formula:
\[ f = \frac{1}{2L} \sqrt{\frac{T}{\mu}} \]
where:
- \( L \) is the length of the string,
- \( T \) is the tension in the string,
- \( \mu \) is the linear density of the string.

### Step 2: Determine the tension for each string
The tension \( T \) in each string is equal to the weight of the mass hanging from it. Therefore:
- For the first string (mass \( m_1 = 12 \, \text{kg} \)):
\[ T_1 = m_1 g = 12g \]
- For the second string (mass \( m_2 = 3 \, \text{kg} \)):
\[ T_2 = m_2 g = 3g \]

### Step 3: Write the frequency equations for both strings
For the first string (fundamental frequency):
\[ f_1 = \frac{1}{2L_1} \sqrt{\frac{12g}{\mu}} \]
For the second string (first overtone):
The first overtone frequency is twice the fundamental frequency:
\[ f_2 = 2 \cdot \frac{1}{2L_2} \sqrt{\frac{3g}{\mu}} = \frac{1}{L_2} \sqrt{\frac{3g}{\mu}} \]

### Step 4: Set the frequencies equal
According to the problem, the fundamental frequency of the first string is equal to the first overtone of the second string:
\[ f_1 = f_2 \]
Substituting the frequency equations:
\[ \frac{1}{2L_1} \sqrt{\frac{12g}{\mu}} = \frac{1}{L_2} \sqrt{\frac{3g}{\mu}} \]

### Step 5: Simplify the equation
We can cancel \( g \) and \( \mu \) from both sides:
\[ \frac{1}{2L_1} \sqrt{12} = \frac{1}{L_2} \sqrt{3} \]
Cross-multiplying gives:
\[ \sqrt{12} L_2 = 2 \sqrt{3} L_1 \]

### Step 6: Substitute values
We know \( L_2 = 100 \, \text{cm} \):
\[ \sqrt{12} \cdot 100 = 2 \sqrt{3} L_1 \]
Calculating \( \sqrt{12} \):
\[ \sqrt{12} = 2\sqrt{3} \]
Substituting this back:
\[ 2\sqrt{3} \cdot 100 = 2 \sqrt{3} L_1 \]
Dividing both sides by \( 2\sqrt{3} \):
\[ 100 = L_1 \]

### Conclusion
The length of the first string \( L_1 \) is \( 100 \, \text{cm} \).

### Final Answer
\[ L_1 = 100 \, \text{cm} \]

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