## Turntable

When a ball is placed on a spinning turntable (rolling with the turntables motion), it follows an unexpected circular path at a 2/7 orbit ratio to the turntable’s rotation.

In Short

When a ball is put on a rotating turntable, common intuition might suggest it would quickly fly off due to centrifugal force. However, something peculiar happens (when the ball is brought up to speed of the turntable). Instead of an early departure, the ball embarks on a mesmerizing circular journey. This behavior is characterized by an intriguing 2/7 orbit ratio, meaning a solid ball completes two orbits for every seven rotations of the turntable. Surprisingly, the ball’s mass, size, and the turntable’s rotation speed seem to have no bearing on this curious phenomenon.

Details and Theory

The key to understanding this peculiar behavior lies in the dynamics of rotational motion. When the ball is placed on the turntable rolling with the motion of the turntable, it initially remains stationary due to the forces of friction between the ball and the turntable’s surface. There is no force pushing it outward, and in the absence of external disturbances, it maintains this stationary position.

However, when nudged or pushed even slightly inwards, an interesting interplay of forces comes into play. The ball starts to roll because its rotational speed exceeds that of the turntable directly beneath it. This rolling motion can be explained using the principles of relative motion. The ball’s absolute velocity is the result of combining its own rotational velocity (relative to its center) and the linear velocity of the turntable’s surface underneath it. These vectors combine to produce a resultant velocity vector, causing the ball to roll.

As the ball progresses along its curved path, its motion aligns with Coriolis force, akin to the Coriolis effect seen in meteorology. This force acts perpendicular to the ball’s velocity, causing its path to curve smoothly. The result is a circular orbit.

Mathematically, the orbital period of the ball can be expressed as a function of the turntable’s orbital period and certain properties of the ball, such as its mass and radius. This ratio, 2/7, emerges as a consequence of this mathematical analysis, and it holds true for solid balls. For hollow balls, the ratio becomes 2/5 due to differences in moment of inertia.

In summary, the 2/7 orbit phenomenon on turntables is an intriguing example, where the ball’s motion and the turntable’s rotation interact in a way that defies common intuition. This phenomenon is a testament to the rich dynamics that can emerge from seemingly simple physical systems.