A

Step 1: Recognize that the particle is moving in a circular path due to the rotation of the groove. The centripetal acceleration required to keep the particle moving in a circular path is given by \( a_c = \frac{v^2}{r} \), where \( v \) is the tangential velocity and \( r \) is the radius of the circular path.
Step 2: Since the groove is rotating with angular velocity \( \omega \), the relationship between linear velocity \( v \) and angular velocity is given by \( v = r \omega \).
Step 3: Initially, at distance \( R \), the initial velocity of the particle is \( v_0 = R \omega \).
Step 4: At a distance \( r \), the velocity of the particle is \( v = r \omega \).
Thus, the expression for the velocity of the particle when it is at a distance \( r \) is \( v = r \omega \).
Therefore, A.