A sonometer wire under tension of 64 N vibrating in its fundamental mode is in resonance with a vibrating tuning fork. The vibrating portion of the sonometer wire has a length of 10 cm and mass of 1 g. The vibrating tuning fork is now moved away from the vibrating wire with a constant speed and an observer standing near the sonometer hears one beat per second. Calculate the speed with which the tuning fork is moved, if the speed of sound in air is 300 m/s.

Options

A. 0.75m/s
B. 0.50m/s
C. 0.30m/s
D. 0.25m/s

Explanation

To solve the problem step by step, we will follow these steps:

### Step 1: Identify the parameters given in the problem
- Tension (T) = 64 N
- Length of the wire (L) = 10 cm = 0.1 m
- Mass of the wire (m) = 1 g = 0.001 kg
- Speed of sound in air (v₀) = 300 m/s
- Beats heard = 1 beat per second

### Step 2: Calculate the mass per unit length (μ) of the wire
The mass per unit length (μ) is given by the formula:
\[ \mu = \frac{m}{L} \]
Substituting the values:
\[ \mu = \frac{0.001 \text{ kg}}{0.1 \text{ m}} = 0.01 \text{ kg/m} \]

### Step 3: Calculate the frequency (f) of the sonometer wire
The frequency of a vibrating wire in its fundamental mode is given by the formula:
\[ f = \frac{1}{2L} \sqrt{\frac{T}{\mu}} \]
Substituting the values:
\[ f = \frac{1}{2 \times 0.1} \sqrt{\frac{64}{0.01}} = \frac{1}{0.2} \sqrt{6400} = 5 \times 80 = 400 \text{ Hz} \]

### Step 4: Determine the apparent frequency when the tuning fork is moved away
Since the observer hears one beat per second, the apparent frequency (f') of the tuning fork when moved away is:
\[ f' = f - 1 = 400 - 1 = 399 \text{ Hz} \]

### Step 5: Apply the Doppler effect formula
Using the Doppler effect for sound, when the source is moving away from a stationary observer, the formula is:
\[ f' = f \frac{v_0}{v_0 + v_s} \]
Where:
- \(f' = 399 \text{ Hz}\)
- \(f = 400 \text{ Hz}\)
- \(v_0 = 300 \text{ m/s}\)
- \(v_s\) is the speed of the tuning fork (what we need to find).

Substituting the known values:
\[ 399 = 400 \frac{300}{300 + v_s} \]

### Step 6: Solve for \(v_s\)
Rearranging the equation:
\[ 399(300 + v_s) = 400 \times 300 \]
Expanding and simplifying:
\[ 119700 + 399v_s = 120000 \]
\[ 399v_s = 120000 - 119700 \]
\[ 399v_s = 300 \]
\[ v_s = \frac{300}{399} \approx 0.752 \text{ m/s} \]

### Final Answer
The speed with which the tuning fork is moved is approximately:
\[ v_s \approx 0.75 \text{ m/s} \]

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