The velocity of transverse wave in string whose linear mass density is `3×10^-2 kg/m` stretched by a load of 30 kg is (Take g=10`m/s^2`)

Options

A. 10m/s
B. 30m/s
C. 300m/s
D. 100m/s

Answer

To find the velocity of a transverse wave in a string, we can use the formula:

\[ V = \sqrt{\frac{T}{\mu}} \]

where:
- \( V \) is the velocity of the wave,
- \( T \) is the tension in the string,
- \( \mu \) is the linear mass density of the string.

### Step 1: Identify the given values
From the question, we have:
- Linear mass density, \( \mu = 3 \times 10^{-2} \, \text{kg/m} \)
- Mass of the load, \( m = 30 \, \text{kg} \)
- Acceleration due to gravity, \( g = 10 \, \text{m/s}^2 \)

### Step 2: Calculate the tension in the string
The tension \( T \) in the string can be calculated using the formula:

\[ T = m \cdot g \]

Substituting the values:

\[ T = 30 \, \text{kg} \times 10 \, \text{m/s}^2 = 300 \, \text{N} \]

### Step 3: Substitute the values into the wave velocity formula
Now, we can substitute the values of \( T \) and \( \mu \) into the wave velocity formula:

\[ V = \sqrt{\frac{T}{\mu}} = \sqrt{\frac{300 \, \text{N}}{3 \times 10^{-2} \, \text{kg/m}}} \]

### Step 4: Simplify the expression
Calculating the denominator:

\[ 3 \times 10^{-2} = 0.03 \, \text{kg/m} \]

Now substituting this back into the equation:

\[ V = \sqrt{\frac{300}{0.03}} = \sqrt{10000} \]

### Step 5: Calculate the final result
Taking the square root:

\[ V = 100 \, \text{m/s} \]

### Final Answer
The velocity of the transverse wave in the string is \( 100 \, \text{m/s} \).

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