Two particles A, B are moving on two concentric circles of radii $$R_1$$ and $$R_2$$ with equal angular speed $$\omega$$. At $$t = 0$$, their positions and direction of motion are shown in the figure. The relative velocity $$\vec{V_A} - \vec{V_B}$$ at $$t = \frac{\pi}{2\omega}$$ is given by:

For the angular displacement of both particles in time $$t = \frac{\pi}{2\omega}$$:

$$\theta = \omega t = \omega \left(\frac{\pi}{2\omega}\right) = \frac{\pi}{2}\text{ rad} = 90^\circ$$

For particle A (initially at $$(R_1, 0)$$ moving in the counter-clockwise direction along $$+\hat{j}$$):

$$\text{After rotating by } 90^\circ\text{, its position is on the positive y-axis, and its velocity points along } -\hat{i}:$$ 

$$\vec{V}_A = -\omega R_1 \hat{i}$$

For particle B (initially at $$(R_2, 0)$$ moving in the clockwise direction along $$-\hat{j}$$):

$$\text{After rotating by } 90^\circ\text{, its position is on the negative y-axis, and its velocity points along } -\hat{i}:$$

$$\vec{V}_B = -\omega R_2 \hat{i}$$

For relative velocity $$\vec{V}_A - \vec{V}_B$$:

$$\vec{V}_A - \vec{V}_B = (-\omega R_1 \hat{i}) - (-\omega R_2 \hat{i}) = \omega(R_2 - R_1)\hat{i}$$