Calculating Repeating Configuration Time

The problem involves a ball rotating internally at a rate '$r$' rotations per second and revolving externally around a stationary point 'O' at a rate '$R$' revolutions per second. We are given that '$R < r$' and both motions occur in the same sense. We need to find the time after which a specific configuration repeats.

Understanding Relative Motion

The configuration described is when a point on the ball lies on the line connecting the ball's center to the stationary point O. This specific alignment depends on the rotation of the ball relative to its revolution around O.

Since the ball rotates at rate '$r$' and revolves at rate '$R$' in the same sense, and '$r > R$', the rotation effectively 'overtakes' the revolution. The relative angular speed of the ball's rotation with respect to its revolution around O is the difference between their rates.

Relative speed = (Rotational speed) - (Revolution speed)

Relative speed = \( r - R \) rotations per second.

Deriving the Time Period

The configuration repeats when the point on the ball completes one full cycle relative to the line segment connecting the ball's center to point O. This corresponds to one full relative rotation.

The time period '$T$' is the time taken for one relative rotation. It is the reciprocal of the relative speed.

Time \( T = \frac{1}{\text{Relative speed}} \)

Substituting the relative speed:

Time \( T = \frac{1}{r-R} \) seconds.