The number of beats produced per second by two vibrations: `x_(1) = x_(0) sin 646 pi t` and `x_(2) = x_(0) sin 652 pi t` is

Options

A. 2
B. 3
C. 4
D. 6

Answer

To find the number of beats produced per second by the two vibrations given by the equations \( x_1 = x_0 \sin(646 \pi t) \) and \( x_2 = x_0 \sin(652 \pi t) \), we can follow these steps:

### Step 1: Identify the angular frequencies
The angular frequency \( \omega \) is related to the frequency \( f \) by the formula:
\[ \omega = 2 \pi f \]
From the equations, we can identify:
- For \( x_1 \): \( \omega_1 = 646 \pi \)
- For \( x_2 \): \( \omega_2 = 652 \pi \)

### Step 2: Calculate the frequencies
Using the relationship between angular frequency and frequency, we can calculate the frequencies \( f_1 \) and \( f_2 \):
\[ f_1 = \frac{\omega_1}{2 \pi} = \frac{646 \pi}{2 \pi} = \frac{646}{2} = 323 \text{ Hz} \]
\[ f_2 = \frac{\omega_2}{2 \pi} = \frac{652 \pi}{2 \pi} = \frac{652}{2} = 326 \text{ Hz} \]

### Step 3: Calculate the beat frequency
The number of beats produced per second is given by the absolute difference in frequencies:
\[ \text{Number of beats} = |f_2 - f_1| = |326 - 323| = 3 \text{ beats per second} \]

### Final Answer
Thus, the number of beats produced per second by the two vibrations is **3 beats per second**.

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