The speed of a transverse wave in a stretched string is 348 `ms^(-1)`, when the tension of the string is 3.6 kg wt. Calculate the speed of the transverse wave in the same string . If the tension in the string is changed to 4.9 kg wt ?

Answer

To solve the problem, we will use the formula for the speed of a transverse wave in a stretched string, which is given by:

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

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

### Step 1: Calculate the linear mass density \( \mu \)

Given:
- Speed of the wave \( V = 348 \, \text{m/s} \)
- Tension \( T = 3.6 \, \text{kg wt} = 3.6 \times 9.81 \, \text{N} = 35.296 \, \text{N} \) (since \( 1 \, \text{kg wt} = 9.81 \, \text{N} \))

Using the formula:

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

Squaring both sides gives:

\[ V^2 = \frac{T}{\mu} \]

Rearranging for \( \mu \):

\[ \mu = \frac{T}{V^2} \]

Substituting the values:

\[ \mu = \frac{35.296}{348^2} \]

Calculating \( 348^2 \):

\[ 348^2 = 121104 \]

Now substituting:

\[ \mu = \frac{35.296}{121104} \approx 0.000291 \, \text{kg/m} \]

### Step 2: Calculate the new speed with the new tension

Now, we need to find the speed of the wave when the tension is changed to \( T = 4.9 \, \text{kg wt} = 4.9 \times 9.81 \, \text{N} = 48.069 \, \text{N} \).

Using the same formula for speed:

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

Substituting the new tension and the previously calculated \( \mu \):

\[ V = \sqrt{\frac{48.069}{0.000291}} \]

Calculating \( \frac{48.069}{0.000291} \):

\[ \frac{48.069}{0.000291} \approx 165,000.34 \]

Now taking the square root:

\[ V \approx \sqrt{165000.34} \approx 406.19 \, \text{m/s} \]

### Final Answer

The speed of the transverse wave in the string when the tension is changed to 4.9 kg wt is approximately **406.19 m/s**.

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