Particle A and B are moving in coplanar circular paths centred at O. They are totating in the same sense. Time periods of rotation of A and B around O are `T_(A)" and "T_(B)`, respectively, with `T_(B)gtT_(A)`. Time required for B to make one ratation around O relative to A is :

Options

A. `T_(B)-T_(A)`
B. `T_(B)+T_(A)`
C. `(T_(B)T_(A))/(T_(B)+T_(A))`
D. `(T_(B)T_(A))/(T_(B)-T_(A))`

Answer

To solve the problem, we need to determine the time required for particle B to make one complete rotation around point O relative to particle A.

### Step-by-Step Solution:

1. **Understanding the Time Periods**:
- Let \( T_A \) be the time period of particle A.
- Let \( T_B \) be the time period of particle B.
- Given that \( T_B > T_A \), this means that particle B takes longer to complete one rotation than particle A.

2. **Relative Motion**:
- We need to find the relative time taken by B to complete one rotation around O as observed from A's perspective.
- The angular velocity \( \omega \) of a particle is given by the formula:
\[ \omega = \frac{2\pi}{T} \]
- Therefore, the angular velocities of particles A and B are:
\[ \omega_A = \frac{2\pi}{T_A} \]
\[ \omega_B = \frac{2\pi}{T_B} \]

3. **Relative Angular Velocity**:
- The relative angular velocity of B with respect to A is given by:
\[ \omega_{BA} = \omega_B - \omega_A \]
- Substituting the values of \( \omega_A \) and \( \omega_B \):
\[ \omega_{BA} = \frac{2\pi}{T_B} - \frac{2\pi}{T_A} \]
- Factoring out \( 2\pi \):
\[ \omega_{BA} = 2\pi \left( \frac{1}{T_B} - \frac{1}{T_A} \right) \]

4. **Time for One Complete Rotation**:
- The time taken for B to complete one rotation relative to A can be found using the formula:
\[ T_{BA} = \frac{2\pi}{\omega_{BA}} \]
- Substituting \( \omega_{BA} \):
\[ T_{BA} = \frac{2\pi}{2\pi \left( \frac{1}{T_B} - \frac{1}{T_A} \right)} = \frac{1}{\left( \frac{1}{T_B} - \frac{1}{T_A} \right)} \]
- Simplifying further:
\[ T_{BA} = \frac{T_A T_B}{T_B - T_A} \]

### Final Answer:
The time required for particle B to make one rotation around O relative to particle A is:
\[ T_{BA} = \frac{T_A T_B}{T_B - T_A} \]