Question

A particle is moving in a uniform circular motion of radius \( R \) with angular velocity \( \omega \) in the anti-clockwise direction. Another particle is also moving along the same circular path but in the clockwise direction with the same angular velocity \( \omega \). Suppose, both the particles start at \( t = 0 \) from the same point (R, 0) in opposite directions. The minimum time \( t \) after which the velocities of the two particles become orthogonal to each other is:

• A. \( \frac{\pi}{4 \omega} \)
• B. \( \frac{\pi}{2 \omega} \)
• C. \( \frac{\pi}{\omega} \)
• D. \( \frac{\pi}{8 \omega} \)

Hint:

When solving problems related to circular motion, pay attention to the directions and angular velocities of the objects. For orthogonality, you can use the angle between the vectors of their velocities.

Answer 1

The Correct Option is A

Given that two particles are moving in uniform circular motion with the same angular velocity \( \omega \) in opposite directions, we can use the following reasoning to find the minimum time after which their velocities become orthogonal:
- Let the two particles start at \( t = 0 \) from the same point. One particle moves anti-clockwise and the other moves clockwise along the same circular path of radius \( R \).
- Initially, both particles have the same angular velocity \( \omega \), but they are moving in opposite directions.
- The velocity of each particle at any given time \( t \) is tangential to the circular path. Since they are moving in opposite directions, at some point their velocities will become orthogonal to each other.
To find the time at which the velocities become orthogonal: - The angle between their velocities will become \( 90^\circ \) (or \( \frac{\pi}{2} \) radians) when they are orthogonal.
- The angular displacement of the particles will be \( \theta_1 = \omega t \) for the first particle and \( \theta_2 = -\omega t \) for the second particle.
- The relative angular displacement between the two particles is \( \theta_1 - \theta_2 = \omega t - (-\omega t) = 2\omega t \).
At the point when their velocities are orthogonal: \[ 2\omega t = \frac{\pi}{2} \] Solving for \( t \): \[ t = \frac{\pi}{4\omega} \] Thus, the minimum time after which the velocities of the two particles become orthogonal is \( \frac{\pi}{4\omega} \).