The tension in the string is changed by 2% what is the change in the transverse wave velocity

Options

A. `1%`
B. `2%`
C. `3%`
D. `4%`

Answer

To solve the problem of how a 2% change in tension affects the transverse wave velocity in a string, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between tension and wave velocity**:
The velocity \( V \) of a transverse 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. **Assume mass per unit length is constant**:
Since the problem states that only the tension changes and not the mass per unit length, we can treat \( \mu \) as a constant.

3. **Express velocity in terms of tension**:
We can express the velocity in terms of tension as:
\[ V \propto T^{1/2} \]
This means that the velocity is proportional to the square root of the tension.

4. **Differentiate to find the change in velocity**:
Taking the natural logarithm of both sides, we get:
\[ \ln V = \ln k + \frac{1}{2} \ln T \]
Differentiating both sides gives:
\[ \frac{dV}{V} = \frac{1}{2} \frac{dT}{T} \]

5. **Convert to percentage change**:
The percentage change in velocity \( \frac{dV}{V} \times 100 \) can be expressed as:
\[ \frac{dV}{V} \times 100 = \frac{1}{2} \frac{dT}{T} \times 100 \]

6. **Substitute the given percentage change in tension**:
Given that the tension is changed by 2%, we have:
\[ \frac{dT}{T} \times 100 = 2\% \]
Substituting this into the equation gives:
\[ \frac{dV}{V} \times 100 = \frac{1}{2} \times 2\% = 1\% \]

7. **Final result**:
Therefore, the percentage change in the transverse wave velocity is:
\[ 1\% \]

### Conclusion:
The change in the transverse wave velocity due to a 2% change in tension is **1%**.