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 25g and length of the 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 of finding the percentage error in the measurement of the velocity of a transverse wave in a string, we will follow these steps:

### Step 1: Understand the Formula
The velocity \( v \) 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: Calculate Mass per Unit Length
The mass per unit length \( m \) can be calculated using the formula:
\[ m = \frac{\text{mass of the string}}{\text{length of the string}} \]
Given that the mass of the string is 25 g (which is 0.025 kg) and the length of the string is 1 m:
\[ m = \frac{0.025 \, \text{kg}}{1 \, \text{m}} = 0.025 \, \text{kg/m} \]

### Step 3: Convert Tension to Newtons
The tension \( T \) is given as 3.0 kgf. To convert this to Newtons, we use the conversion factor \( 1 \, \text{kgf} = 9.81 \, \text{N} \):
\[ T = 3.0 \, \text{kgf} \times 9.81 \, \text{N/kgf} = 29.43 \, \text{N} \]

### Step 4: Calculate Velocity
Now we can calculate the velocity \( v \):
\[ v = \sqrt{\frac{T}{m}} = \sqrt{\frac{29.43 \, \text{N}}{0.025 \, \text{kg/m}}} = \sqrt{1177.2} \approx 34.3 \, \text{m/s} \]

### Step 5: Determine the Errors
Next, we need to determine the errors in the measurements:
- For tension \( T \), the least count is 0.1 kgf, which is:
\[ \Delta T = 0.1 \, \text{kgf} \times 9.81 \, \text{N/kgf} = 0.981 \, \text{N} \]
- For length \( L \), the least count is 0.001 m:
\[ \Delta L = 0.001 \, \text{m} \]
- For mass \( m \), the least count is 0.1 g, which is:
\[ \Delta m = 0.1 \, \text{g} \times 0.001 \, \text{kg/g} = 0.0001 \, \text{kg} \]

### Step 6: Calculate Relative Errors
Now we can calculate the relative errors:
\[ \frac{\Delta T}{T} = \frac{0.981}{29.43} \approx 0.0333 \]
\[ \frac{\Delta L}{L} = \frac{0.001}{1} = 0.001 \]
\[ \frac{\Delta m}{m} = \frac{0.0001}{0.025} = 0.004 \]

### Step 7: Combine the Errors
Using the formula for the propagation of uncertainty:
\[ \frac{\Delta v}{v} = \frac{1}{2} \left( \frac{\Delta T}{T} + \frac{\Delta L}{L} + \frac{\Delta m}{m} \right) \]
Substituting the values:
\[ \frac{\Delta v}{v} = \frac{1}{2} \left( 0.0333 + 0.001 + 0.004 \right) = \frac{1}{2} \times 0.0383 \approx 0.01915 \]

### Step 8: Calculate Percentage Error
To find the percentage error:
\[ \text{Percentage Error} = \frac{\Delta v}{v} \times 100 \approx 0.01915 \times 100 \approx 1.915\% \]

### Final Answer
The percentage error in the measurement of velocity is approximately **1.92%**.

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