A sonometer wire of length 1 m is stretched by a weight of 10 kg. The fundamental frequency of vibration is 100 Hz. Determine the linear density of material of wire.

Answer

To determine the linear density of the sonometer wire, we can follow these steps:

### Step 1: Identify the given values
- Length of the wire (L) = 1 m
- Weight (mass) (m) = 10 kg
- Fundamental frequency (f₀) = 100 Hz
- Acceleration due to gravity (g) = 9.8 m/s² (approximately)

### Step 2: Calculate the tension (T) in the wire
The tension in the wire is equal to the weight of the mass hanging from it. This can be calculated using the formula:
\[ T = m \cdot g \]
Substituting the values:
\[ T = 10 \, \text{kg} \cdot 9.8 \, \text{m/s}^2 = 98 \, \text{N} \]

### Step 3: Use the formula for fundamental frequency
The formula for the fundamental frequency of a sonometer wire is given by:
\[ f₀ = \frac{1}{2L} \sqrt{\frac{T}{\mu}} \]
Where:
- \( f₀ \) = fundamental frequency
- \( L \) = length of the wire
- \( T \) = tension in the wire
- \( \mu \) = linear density of the wire

### Step 4: Rearrange the formula to solve for linear density (μ)
Rearranging the formula to isolate μ:
\[ f₀ = \frac{1}{2L} \sqrt{\frac{T}{\mu}} \]
Squaring both sides gives:
\[ f₀^2 = \left(\frac{1}{2L}\right)^2 \frac{T}{\mu} \]
Rearranging further:
\[ \mu = \frac{T}{(2Lf₀)^2} \]

### Step 5: Substitute the known values into the equation
Now substituting the values we have:
- \( T = 98 \, \text{N} \)
- \( L = 1 \, \text{m} \)
- \( f₀ = 100 \, \text{Hz} \)

Calculating:
\[ \mu = \frac{98}{(2 \cdot 1 \cdot 100)^2} \]
\[ \mu = \frac{98}{(200)^2} \]
\[ \mu = \frac{98}{40000} \]
\[ \mu = 0.00245 \, \text{kg/m} \]
Or in scientific notation:
\[ \mu = 2.45 \times 10^{-3} \, \text{kg/m} \]

### Final Answer
The linear density of the material of the wire is \( 2.45 \times 10^{-3} \, \text{kg/m} \).

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