A steel wire of length `64 cm` weighs `5 g`. If it is stretched by a force of `8 `N` what would be the speed of transverse wave passing on it?

Explanation

To find the speed of the transverse wave passing through a steel wire, we can follow these steps:

### Step 1: Convert the given measurements to SI units
- Length of the wire, \( L = 64 \, \text{cm} = 64 \times 10^{-2} \, \text{m} = 0.64 \, \text{m} \)
- Mass of the wire, \( m = 5 \, \text{g} = 5 \times 10^{-3} \, \text{kg} = 0.005 \, \text{kg} \)

### Step 2: Calculate the linear mass density (\( \mu \))
The linear mass density \( \mu \) is given by the formula:
\[ \mu = \frac{m}{L} \]
Substituting the values:
\[ \mu = \frac{0.005 \, \text{kg}}{0.64 \, \text{m}} = \frac{5 \times 10^{-3}}{0.64} \approx 0.0078125 \, \text{kg/m} \]

### Step 3: Identify the tension in the wire
The tension \( T \) in the wire is given as:
\[ T = 8 \, \text{N} \]

### Step 4: Use the formula for the speed of a wave in a stretched string
The speed \( v \) of a transverse wave in a string is given by the formula:
\[ v = \sqrt{\frac{T}{\mu}} \]
Substituting the values of \( T \) and \( \mu \):
\[ v = \sqrt{\frac{8 \, \text{N}}{0.0078125 \, \text{kg/m}}} \]

### Step 5: Calculate the speed
Calculating the value inside the square root:
\[ v = \sqrt{1024} = 32 \, \text{m/s} \]

### Final Answer
The speed of the transverse wave passing through the steel wire is \( 32 \, \text{m/s} \).

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