In an experiment it was found that a sonometer in its fundamental mode of vibration and a tuning fork gave 5 beats when length of wire is 1.05 metre or 1 metre. The velocity of tranverse waves in sonometer wire when its length is 1m

Options

A. 400 m/s
B. 210 m/s
C. 420 m/s
D. 450 m/s

Answer

To solve the problem, let's follow these steps:

### Step 1: Understand the problem
We know that a sonometer wire in its fundamental mode of vibration produces beats with a tuning fork. The lengths of the wire are given as 1.05 meters and 1 meter, and the number of beats observed is 5. We need to find the velocity of transverse waves in the sonometer wire when its length is 1 meter.

### Step 2: Use the formula for frequency
The fundamental frequency \( f \) of a vibrating string is given by the formula:
\[ f = \frac{V}{2L} \]
where \( V \) is the velocity of the wave in the string and \( L \) is the length of the string.

### Step 3: Set up the frequency difference
Let \( f_1 \) be the frequency of the wire at length \( L_1 = 1.05 \, \text{m} \) and \( f_2 \) be the frequency at length \( L_2 = 1 \, \text{m} \).

Using the frequency formula:
\[ f_1 = \frac{V}{2 \times 1.05} \]
\[ f_2 = \frac{V}{2 \times 1} \]

### Step 4: Calculate the difference in frequencies
The difference in frequencies is given by:
\[ |f_1 - f_2| = 5 \text{ beats} \]

Since we have 5 beats, the difference in frequencies can be expressed as:
\[ |f_1 - f_2| = 10 \text{ (because each beat corresponds to a frequency difference of 1 Hz)} \]

### Step 5: Substitute the frequencies into the equation
Substituting \( f_1 \) and \( f_2 \) into the equation gives:
\[ \left| \frac{V}{2 \times 1.05} - \frac{V}{2 \times 1} \right| = 10 \]

### Step 6: Simplify the equation
Taking \( V \) common:
\[ \left| \frac{V}{2} \left( \frac{1}{1.05} - 1 \right) \right| = 10 \]
This simplifies to:
\[ \left| \frac{V}{2} \left( \frac{1 - 1.05}{1.05} \right) \right| = 10 \]
\[ \left| \frac{V}{2} \left( \frac{-0.05}{1.05} \right) \right| = 10 \]

### Step 7: Remove the absolute value
Since we are dealing with a positive velocity, we can drop the absolute value:
\[ \frac{V}{2} \cdot \frac{0.05}{1.05} = 10 \]

### Step 8: Solve for \( V \)
Rearranging gives:
\[ V = 10 \cdot 2 \cdot \frac{1.05}{0.05} \]
Calculating this:
\[ V = 20 \cdot 21 = 420 \, \text{m/s} \]

### Conclusion
The velocity of transverse waves in the sonometer wire when its length is 1 meter is:
\[ \boxed{420 \, \text{m/s}} \]