A uniform wire of length 20 m and weighing 5 kg hangs vertically. If `g=10 m s^(-2)`, then the speed of transverse waves in the middle of the wire is

Options

A. `10 m s^(-1)`
B. `10sqrt2 m s^(-1)`
C. `4 m s^(-1)`
D. `2 m s^(-1)`

Answer

To find the speed of transverse waves in the middle of a uniform wire, we can follow these steps:

### Step 1: Determine the Tension in the Wire
The tension \( T \) in the middle of the wire can be calculated using the formula:
\[ T = \frac{m}{2} \cdot g \]
where:
- \( m \) is the total mass of the wire (5 kg),
- \( g \) is the acceleration due to gravity (10 m/s²).

Substituting the values:
\[ T = \frac{5 \, \text{kg}}{2} \cdot 10 \, \text{m/s}^2 = \frac{5 \cdot 10}{2} = 25 \, \text{N} \]

### Step 2: Calculate the Mass per Unit Length (\( \mu \))
The mass per unit length \( \mu \) is given by:
\[ \mu = \frac{m}{L} \]
where:
- \( L \) is the length of the wire (20 m).

Substituting the values:
\[ \mu = \frac{5 \, \text{kg}}{20 \, \text{m}} = \frac{1}{4} \, \text{kg/m} \]

### Step 3: Calculate the Speed of Transverse Waves
The speed \( v \) of transverse waves in the wire can be calculated using the formula:
\[ v = \sqrt{\frac{T}{\mu}} \]
Substituting the values of \( T \) and \( \mu \):
\[ v = \sqrt{\frac{25 \, \text{N}}{\frac{1}{4} \, \text{kg/m}}} = \sqrt{25 \cdot 4} = \sqrt{100} = 10 \, \text{m/s} \]

### Final Answer
The speed of transverse waves in the middle of the wire is \( 10 \, \text{m/s} \).

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