If two particles are moving on same circle in same plane with different angular velocities `omega_(1)` and `omega_(2)` and different time period `T_(1)`, and `T_(2)` in same sense, then the time taken by 2 to complete one revolution w.r.t particle l is

Options

A. `T= (T_(1)T_(2))/(T_(2)-T_(1))`
B. `T= (T_(1)+T_(2))/(2)`
C. `T= (T_(1)T_(2))/(T_(1)+T_(2))`
D. `T_(2)-T_(1)`

Answer

To solve the problem, we need to find the time taken by particle 2 to complete one revolution with respect to particle 1. Let's break down the solution step by step.

### Step-by-Step Solution:

1. **Understanding Angular Velocity and Time Period**:
- The angular velocity \( \omega \) and time period \( T \) are related by the formula:
\[ T = \frac{2\pi}{\omega} \]
- For particle 1, we have:
\[ T_1 = \frac{2\pi}{\omega_1} \]
- For particle 2, we have:
\[ T_2 = \frac{2\pi}{\omega_2} \]

2. **Finding Relative Angular Velocity**:
- Since both particles are moving in the same direction, the relative angular velocity \( \omega_{\text{relative}} \) between the two particles is given by:
\[ \omega_{\text{relative}} = \omega_1 - \omega_2 \]

3. **Expressing Relative Angular Velocity in Terms of Time Periods**:
- We can express \( \omega_1 \) and \( \omega_2 \) in terms of \( T_1 \) and \( T_2 \):
\[ \omega_1 = \frac{2\pi}{T_1}, \quad \omega_2 = \frac{2\pi}{T_2} \]
- Substituting these into the relative angular velocity equation gives:
\[ \omega_{\text{relative}} = \frac{2\pi}{T_1} - \frac{2\pi}{T_2} \]

4. **Finding the Time Taken for One Revolution Relative to Particle 1**:
- The time taken for particle 2 to complete one revolution with respect to particle 1 is given by:
\[ T_{\text{relative}} = \frac{2\pi}{\omega_{\text{relative}}} \]
- Substituting the expression for \( \omega_{\text{relative}} \):
\[ T_{\text{relative}} = \frac{2\pi}{\left(\frac{2\pi}{T_1} - \frac{2\pi}{T_2}\right)} \]
- Simplifying this expression:
\[ T_{\text{relative}} = \frac{2\pi}{2\pi\left(\frac{1}{T_1} - \frac{1}{T_2}\right)} = \frac{1}{\left(\frac{1}{T_1} - \frac{1}{T_2}\right)} \]
- This can be rewritten as:
\[ T_{\text{relative}} = \frac{T_1 T_2}{T_2 - T_1} \]

5. **Final Result**:
- The time taken by particle 2 to complete one revolution with respect to particle 1 is:
\[ T_{\text{relative}} = \frac{T_1 T_2}{T_2 - T_1} \]