A sonometer wire supports a 4 kg load and vibrates in fundamental mode with a tunig fork of frequency 416 Hz. The length of the wire between the bridges is now doubled. In order to maintain fundamental mode, the load should be changed to

Options

A. 1 kg
B. 2 kg
C. 8 kg
D. 16 kg

Answer

To solve the problem step by step, we will use the relationship between the fundamental frequency of a vibrating string, its length, tension, and mass per unit length.

### Step 1: Understand the fundamental frequency formula
The fundamental frequency \( f \) of a vibrating string is given by the formula:
\[ f = \frac{1}{2L} \sqrt{\frac{T}{\mu}} \]
where:
- \( L \) = length of the string
- \( T \) = tension in the string
- \( \mu \) = mass per unit length of the string

### Step 2: Calculate the initial conditions
In the initial case, we have:
- Load \( m_1 = 4 \, \text{kg} \)
- Tension \( T_1 = m_1 \cdot g = 4 \cdot 10 = 40 \, \text{N} \) (taking \( g \approx 10 \, \text{m/s}^2 \))
- Length \( L_1 \)

Using the fundamental frequency:
\[ f = \frac{1}{2L_1} \sqrt{\frac{40}{\mu}} \]

### Step 3: Set up the equation for the new conditions
When the length of the wire is doubled, the new length \( L_2 = 2L_1 \). We want to maintain the same fundamental frequency \( f \).

For the new conditions, let the new mass be \( m_2 \) and the new tension be:
\[ T_2 = m_2 \cdot g \]

The equation for the new fundamental frequency becomes:
\[ f = \frac{1}{2L_2} \sqrt{\frac{T_2}{\mu}} = \frac{1}{4L_1} \sqrt{\frac{m_2 \cdot g}{\mu}} \]

### Step 4: Equate the two frequencies
Since we want the frequencies to remain the same:
\[ \frac{1}{2L_1} \sqrt{\frac{40}{\mu}} = \frac{1}{4L_1} \sqrt{\frac{m_2 \cdot g}{\mu}} \]

### Step 5: Simplify the equation
Cancel \( L_1 \) and \( \mu \) from both sides:
\[ \frac{1}{2} \sqrt{40} = \frac{1}{4} \sqrt{m_2 \cdot g} \]

### Step 6: Cross-multiply and square both sides
Cross-multiplying gives:
\[ 2 \sqrt{40} = \sqrt{m_2 \cdot g} \]

Squaring both sides results in:
\[ 4 \cdot 40 = m_2 \cdot g \]
\[ 160 = m_2 \cdot g \]

### Step 7: Solve for \( m_2 \)
Now, substituting \( g \approx 10 \, \text{m/s}^2 \):
\[ m_2 = \frac{160}{10} = 16 \, \text{kg} \]

### Conclusion
To maintain the fundamental mode after doubling the length of the wire, the load should be changed to **16 kg**.

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