Two particles move on a circular path (one just inside and the other just outside) with the angular velocities `omega` and `5omega` starting from the same point. Then

Options

A. the cross each other at regular intervals of time `2pi//4omega` when their angular velocities are oppositely directed
B. they cross each other at point on the path subtending an angle of `60^@` at the centre if their angular velocities are oppositely directed
C. the cross at intervals of time `pi//3omega` if their angular velocities are oppositely directed
D. they cross each other at points on the path subtendingf `90^` at the centre if their angular velocities are in the same sence

Explanation

To solve the problem, we need to analyze the motion of two particles moving on a circular path with different angular velocities. Let's break down the solution step by step.

### Step 1: Define the Problem
We have two particles moving on a circular path:
- Particle A moves with angular velocity \( \omega \).
- Particle B moves with angular velocity \( 5\omega \).
They start from the same point on the circle.

### Step 2: Determine the Time of Intersection
Since they are moving in opposite directions, we can set up the equations for their angular positions at time \( t \):
- For Particle A: \( \theta_A = \omega t \)
- For Particle B: \( \theta_B = 2\pi - 5\omega t \)

### Step 3: Set Up the Equation for Intersection
The particles will cross each other when their angular positions are equal:
\[ \omega t = 2\pi - 5\omega t \]
Rearranging gives:
\[ \omega t + 5\omega t = 2\pi \]
\[ 6\omega t = 2\pi \]
\[ t = \frac{2\pi}{6\omega} = \frac{\pi}{3\omega} \]

### Step 4: Calculate the Angle Subtended
Now, we can find the angle \( \theta \) subtended at the center by Particle A when they meet:
\[ \theta = \omega t = \omega \left(\frac{\pi}{3\omega}\right) = \frac{\pi}{3} \]
This means \( \theta = 60^\circ \).

### Step 5: Analyze the Options
Now we can analyze the options given in the problem:
1. They cross each other at regular intervals of time \( \frac{2\pi}{4\omega} \) - **Not Correct**.
2. They cross each other at points on the path subtending an angle \( \theta = 60^\circ \) - **Correct**.
3. They cross at intervals of time \( \frac{\pi}{3\omega} \) - **Correct**.
4. They cross each other at points on the path subtending an angle \( 90^\circ \) when moving in the same sense - **Correct**.

### Conclusion
The correct options are B, C, and D.