Find beat frequency if the motion of two particles is given by
`y_1 = 0.25sin(310 t)`
`y_2 =0.25 sin (316t)`

Options

A. (a)3
B. (b)`3/pi`
C. (c)`6/pi`
D. (d)6

Answer

To find the beat frequency of the motion of two particles given by the equations \( y_1 = 0.25 \sin(310t) \) and \( y_2 = 0.25 \sin(316t) \), we can follow these steps:

### Step 1: Identify the angular frequencies
From the equations, we can identify the angular frequencies:
- For \( y_1 \): \( \omega_1 = 310 \, \text{rad/s} \)
- For \( y_2 \): \( \omega_2 = 316 \, \text{rad/s} \)

### Step 2: Convert angular frequencies to frequencies
The relationship between angular frequency (\( \omega \)) and frequency (\( f \)) is given by:
\[ \omega = 2\pi f \]
Thus, we can find the frequencies \( f_1 \) and \( f_2 \) using the formula:
\[ f = \frac{\omega}{2\pi} \]

Calculating \( f_1 \):
\[ f_1 = \frac{310}{2\pi} = \frac{155}{\pi} \, \text{Hz} \]

Calculating \( f_2 \):
\[ f_2 = \frac{316}{2\pi} = \frac{158}{\pi} \, \text{Hz} \]

### Step 3: Calculate the beat frequency
The beat frequency (\( f_b \)) is given by the absolute difference between the two frequencies:
\[ f_b = |f_1 - f_2| \]
Substituting the values we found:
\[ f_b = \left| \frac{155}{\pi} - \frac{158}{\pi} \right| = \left| \frac{155 - 158}{\pi} \right| = \left| \frac{-3}{\pi} \right| = \frac{3}{\pi} \, \text{Hz} \]

### Final Answer
The beat frequency is:
\[ f_b = \frac{3}{\pi} \, \text{Hz} \]