The velocity of transverse wave in a string is `v = sqrt( T//m)` where `T` is the tension in the string and `m` is the mass per unit length . If `T = 3.0 kgf`, the mass of string is `v = 1.000 m `, then the percentage error in the measurement of velocity is

Options

A. `0.5`
B. `0.7`
C. `2.3`
D. `3.6`

Explanation

To solve the problem, we will follow these steps:

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

### Step 2: Identify the values given in the problem
From the problem, we have:
- Tension \( T = 3.0 \, \text{kgf} \)
- Length of the string \( L = 1.000 \, \text{m} \)
- Mass per unit length \( m = \frac{\text{mass}}{\text{length}} \)

### Step 3: Convert tension from kgf to Newtons
To convert kgf to Newtons, we use the relation:
\[ 1 \, \text{kgf} = 9.81 \, \text{N} \]
Thus,
\[ T = 3.0 \, \text{kgf} \times 9.81 \, \text{N/kgf} = 29.43 \, \text{N} \]

### Step 4: Calculate mass per unit length
Assuming the mass of the string is \( m' \), we can express mass per unit length as:
\[ m = \frac{m'}{L} \]
We need to find \( m' \) to calculate \( m \).

### Step 5: Substitute values into the velocity formula
Now substituting \( T \) and \( m \) into the velocity formula:
\[ v = \sqrt{\frac{T}{m}} = \sqrt{\frac{29.43}{m}} \]

### Step 6: Calculate the percentage error in velocity
To find the percentage error in the measurement of velocity, we use the formula:
\[ \frac{\Delta v}{v} = \frac{1}{2} \left( \frac{\Delta T}{T} + \frac{\Delta m}{m} + \frac{\Delta L}{L} \right) \]
Where:
- \( \Delta T \) is the error in tension
- \( \Delta m \) is the error in mass
- \( \Delta L \) is the error in length

### Step 7: Determine the least counts
Assuming:
- The least count of \( T \) (tension) is \( 0.1 \, \text{kgf} \)
- The least count of \( L \) (length) is \( 0.001 \, \text{m} \)
- The mass is \( 2.5 \, \text{kg} \) (assumed for calculation)

### Step 8: Substitute the errors into the formula
Substituting the values into the error formula:
- \( \Delta T = 0.1 \, \text{kgf} = 0.1 \times 9.81 \, \text{N} = 0.981 \, \text{N} \)
- \( \Delta m = 0.1 \, \text{kg} \)
- \( \Delta L = 0.001 \, \text{m} \)

Now substituting:
\[ \frac{\Delta v}{v} = \frac{1}{2} \left( \frac{0.981}{29.43} + \frac{0.1}{2.5} + \frac{0.001}{1.000} \right) \]

### Step 9: Calculate each term
Calculating each term:
- \( \frac{0.981}{29.43} \approx 0.0333 \)
- \( \frac{0.1}{2.5} = 0.04 \)
- \( \frac{0.001}{1.000} = 0.001 \)

### Step 10: Sum the terms
Now summing these up:
\[ \frac{\Delta v}{v} \approx \frac{1}{2} (0.0333 + 0.04 + 0.001) = \frac{1}{2} (0.0743) \approx 0.03715 \]

### Step 11: Calculate the percentage error
Finally, the percentage error is:
\[ \text{Percentage Error} = \frac{\Delta v}{v} \times 100 \approx 0.03715 \times 100 \approx 3.715\% \]

### Conclusion
Thus, the percentage error in the measurement of velocity is approximately **3.7%**.

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