If tension is increased by `4%` in vibrating string find `% ` change in speed of wave?

Answer

To solve the problem of finding the percentage change in the speed of a wave when the tension in a vibrating string is increased by 4%, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the formula for wave speed**:
The speed \( v \) of a wave on a 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. **Determine the new tension**:
If the tension is increased by 4%, the new tension \( T' \) can be expressed as:
\[ T' = T + 0.04T = 1.04T \]

3. **Calculate the new wave speed**:
The new speed \( v' \) with the increased tension can be calculated as:
\[ v' = \sqrt{\frac{T'}{\mu}} = \sqrt{\frac{1.04T}{\mu}} = \sqrt{1.04} \cdot \sqrt{\frac{T}{\mu}} = \sqrt{1.04} \cdot v \]

4. **Find the ratio of new speed to old speed**:
The ratio of the new speed to the old speed is:
\[ \frac{v'}{v} = \sqrt{1.04} \]

5. **Calculate the percentage change in speed**:
To find the percentage change in speed, we can use the formula:
\[ \text{Percentage Change} = \left( \frac{v' - v}{v} \right) \times 100 \]
Substituting \( v' = \sqrt{1.04} \cdot v \):
\[ \text{Percentage Change} = \left( \sqrt{1.04} - 1 \right) \times 100 \]

6. **Calculate \( \sqrt{1.04} \)**:
Using a calculator, we find:
\[ \sqrt{1.04} \approx 1.0198 \]
Therefore:
\[ \text{Percentage Change} = (1.0198 - 1) \times 100 \approx 1.98\% \]

7. **Round the result**:
Rounding 1.98% gives us approximately 2%.

### Final Answer:
The percentage change in the speed of the wave is approximately **2%**.

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