Ball Rotation and Revolution

The problem describes a ball that is simultaneously rotating about its own center and revolving around a stationary point. Both motions occur in the same direction (sense). We are asked to find the time after which a specific point on the ball, which is initially in the line connecting the stationary point O and the center of the ball C, repeats this configuration.

Angular Speeds

Let the rate of rotation of the ball be $r$ rotations per second. The angular speed of rotation, $\omega_r$, is related to the rotation rate by: $$ \omega_r = 2\pi r \text{ radians per second} $$ Let the rate of revolution of the ball's center around the stationary point O be $R$ revolutions per second. The angular speed of revolution, $\omega_R$, is related to the revolution rate by: $$ \omega_R = 2\pi R \text{ radians per second} $$ Both the rotation and revolution are in the same direction.

Relative Angular Speed

Consider a specific point P on the surface of the ball. This point P rotates about the center of the ball C. From the perspective of a fixed point in space, the line segment CP rotates with an angular speed $\omega_r$. The center of the ball C revolves around the stationary point O. The line segment OC (connecting O and C) revolves around O with an angular speed $\omega_R$. Initially, the point P is in the line of O and C. Let's assume P is on the side of C away from O (O-C-P). For this configuration to repeat, the point P must again be in the line OC in the same relative position. This means the line segment CP must align with the line segment OC in the same relative orientation.

We can analyze the motion of the line segment CP relative to the line segment OC. Since both are rotating/revolving in the same direction, and the rotation rate $r$ is greater than the revolution rate $R$ (given as $R < r$), the line segment CP is rotating faster than the line segment OC is revolving. The relative angular speed of the line segment CP with respect to the line segment OC is the difference between their absolute angular speeds: $$ \omega_{rel} = \omega_r - \omega_R $$

Time for Configuration to Repeat

For the configuration (point P being on the line OC in the same relative position) to repeat, the line segment CP must complete a full rotation (or an integer multiple of full rotations) relative to the line segment OC. The smallest time for the configuration to repeat is when the relative angle covered is $2\pi$ radians.

Let $T$ be the time taken for the configuration to repeat. The relative angle covered in time $T$ is $\omega_{rel} T$. For repetition, this relative angle must be $2\pi$: $$ \omega_{rel} T = 2\pi $$ Substitute $\omega_{rel} = \omega_r - \omega_R$: $$ (\omega_r - \omega_R) T = 2\pi $$ Now substitute the expressions for $\omega_r$ and $\omega_R$ in terms of rates $r$ and $R$: $$ (2\pi r - 2\pi R) T = 2\pi $$ Factor out $2\pi$: $$ 2\pi (r - R) T = 2\pi $$ Divide both sides by $2\pi$: $$ (r - R) T = 1 $$ Solve for $T$: $$ T = \frac{1}{r-R} $$

This is the time required for the specific point on the ball to be back in the line of the center of the ball and point O, repeating the initial configuration.