A tuning fork of frequency `256 h_(Z)` is moving towards a well with a velocity of `5 m//s` . If the speed of sound is `330 m//s` , then the number of beats heard per second by a stationary observer lying between tuning fork and the well is

Options

A. `2`
B. `4`
C. zero
D. `8`

Explanation

To solve the problem step by step, we will use the concept of the Doppler effect to find the frequency heard by the observer and then determine the number of beats.

### Step 1: Identify the given values
- Frequency of the tuning fork (source), \( f = 256 \, \text{Hz} \)
- Velocity of the tuning fork (source), \( v_s = 5 \, \text{m/s} \) (moving towards the wall)
- Speed of sound in air, \( v = 330 \, \text{m/s} \)

### Step 2: Calculate the frequency heard by the observer (Doppler effect)
Since the tuning fork is moving towards the wall and the observer is stationary, we can use the Doppler effect formula for a moving source and a stationary observer:

\[ f' = f \left( \frac{v}{v - v_s} \right) \]

Substituting the values:

\[ f' = 256 \left( \frac{330}{330 - 5} \right) \]

Calculating the denominator:

\[ 330 - 5 = 325 \]

Now substituting back:

\[ f' = 256 \left( \frac{330}{325} \right) \]

Calculating \( \frac{330}{325} \):

\[ \frac{330}{325} \approx 1.0154 \]

Now calculating \( f' \):

\[ f' \approx 256 \times 1.0154 \approx 260 \, \text{Hz} \]

### Step 3: Frequency of the sound reflected back from the wall
The wall acts as a stationary observer reflecting the sound back. The frequency of the sound reflected back to the observer is the same as the frequency heard by the wall:

\[ f'' = f' = 260 \, \text{Hz} \]

### Step 4: Calculate the beats heard by the observer
The number of beats heard by the observer is the difference between the frequency of the sound heard by the observer and the frequency of the sound reflected back from the wall. Since both frequencies are equal:

\[ \text{Beats} = |f' - f''| = |260 - 260| = 0 \, \text{Hz} \]

### Final Answer
The number of beats heard per second by the stationary observer is \( 0 \, \text{Hz} \).

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