The percentage increase in the speed of transverse waves produced in a stretched string if the tension is increased by 4%, will be _____________%.

Answer

To find the percentage increase in the speed of transverse waves produced in a stretched string when the tension is increased by 4%, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the formula for wave speed**: The speed \( v \) of transverse waves in a stretched string is given by the formula:
\[ v = \sqrt{\frac{T}{\mu}} \]
where \( T \) is the tension in the string and \( \mu \) is the mass per unit length of the string.

2. **Identify the change in tension**: We are given that the tension \( T \) is increased by 4%. This means:
\[ \Delta T = 0.04T \]

3. **Calculate the new tension**: The new tension \( T' \) can be expressed as:
\[ T' = T + \Delta T = T + 0.04T = 1.04T \]

4. **Determine the new speed**: The new speed \( v' \) with the increased tension becomes:
\[ v' = \sqrt{\frac{T'}{\mu}} = \sqrt{\frac{1.04T}{\mu}} = \sqrt{1.04} \cdot \sqrt{\frac{T}{\mu}} = \sqrt{1.04} \cdot v \]

5. **Calculate the percentage increase in speed**: To find the percentage increase in speed, we first find \( \sqrt{1.04} \):
\[ \sqrt{1.04} \approx 1.02 \]
This indicates that the new speed \( v' \) is approximately 2% greater than the original speed \( v \).

6. **Final calculation**: The percentage increase in speed is given by:
\[ \text{Percentage Increase} = \left( \frac{v' - v}{v} \right) \times 100 = \left( \sqrt{1.04} - 1 \right) \times 100 \approx (1.02 - 1) \times 100 = 2\% \]

### Conclusion:
The percentage increase in the speed of transverse waves produced in a stretched string when the tension is increased by 4% is **2%**.

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