Two vibrating tuning forks producing waves given by `y_(1) = 27 "sin" 600 pi t "and" y_(2) = 27 "sin" 604 pi t` are held near the ear of a person, how many beats will be heard in three seconds by him ?

Options

A. 4
B. 2
C. 6
D. 12

Answer

To solve the problem of how many beats will be heard in three seconds from the two tuning forks, we will follow these steps:

### Step 1: Identify the wave equations
The wave equations given are:
- \( y_1 = 27 \sin(600 \pi t) \)
- \( y_2 = 27 \sin(604 \pi t) \)

### Step 2: Determine the angular frequencies
The angular frequency \( \omega \) is related to the frequency \( f \) by the equation:
\[ \omega = 2\pi f \]

From the wave equations:
- For \( y_1 \), \( \omega_1 = 600 \pi \)
- For \( y_2 \), \( \omega_2 = 604 \pi \)

### Step 3: Calculate the frequencies
Using the relation \( f = \frac{\omega}{2\pi} \):
- For \( y_1 \):
\[ f_1 = \frac{600 \pi}{2\pi} = 300 \text{ Hz} \]
- For \( y_2 \):
\[ f_2 = \frac{604 \pi}{2\pi} = 302 \text{ Hz} \]

### Step 4: Calculate the beat frequency
The beat frequency \( f_{\text{beat}} \) is given by the absolute difference between the two frequencies:
\[ f_{\text{beat}} = |f_1 - f_2| = |300 - 302| = 2 \text{ Hz} \]

### Step 5: Calculate the total beats in three seconds
Since the beat frequency is 2 beats per second, the total number of beats in three seconds is:
\[ \text{Total beats} = f_{\text{beat}} \times \text{time} = 2 \text{ Hz} \times 3 \text{ s} = 6 \text{ beats} \]

### Final Answer
The person will hear **6 beats** in three seconds.

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