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. 3
B. `3/pi`
C. `6/pi`
D. 6

Answer

To find the beat frequency of the 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
The angular frequency \( \omega \) is the coefficient of \( t \) in the sine function.

For \( y_1 \):
\[ \omega_1 = 310 \, \text{rad/s} \]

For \( y_2 \):
\[ \omega_2 = 316 \, \text{rad/s} \]

### Step 2: Calculate the frequencies
The frequency \( f \) can be calculated from the angular frequency using the formula:
\[ f = \frac{\omega}{2\pi} \]

For \( y_1 \):
\[ f_1 = \frac{310}{2\pi} \approx \frac{310}{6.2832} \approx 49.4 \, \text{Hz} \]

For \( y_2 \):
\[ f_2 = \frac{316}{2\pi} \approx \frac{316}{6.2832} \approx 50.3 \, \text{Hz} \]

### Step 3: Calculate the beat frequency
The beat frequency \( F_b \) is given by the absolute difference between the two frequencies:
\[ F_b = |f_2 - f_1| \]
Substituting the values we calculated:
\[ F_b = |50.3 - 49.4| = 0.9 \, \text{Hz} \]

### Final Answer
The beat frequency is:
\[ F_b \approx 0.9 \, \text{Hz} \]

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