Ball collides with spinning ring. Spooky.

[techmologist]

I'm trying to figure out what is the resulting motion when a ball collides with a spinning ring whose center of mass is initially at rest. See attached diagrams. The collision is assumed to be elastic. The ball has unit mass, and the ring has unit mass and unit radius.

Shown in the "before impact" diagram are the unit vectors of the coordinate system. The initial angular velocity $$\bold{\omega_i}$$ of the ring is along the x_1 axis. The initial velocity u of the ball is also in the x_1 direction: it is aimed to hit the ring at (0,0,1).

The ring has moment of inertia I = mR^2 = 1 (unit radius and mass) about the x_1 axis, and moment of inertia 1/2 about the x_2 and x_3 axes.

Conservation of energy, momentum, and angular momentum allow me to solve for the final velocity w of the ball, the final velocity v of the ring's COM, and the final angular velocity $$\bold{\omega}$$ of the ring. These are shown in the "immediately after impact" diagram. Using the principal moments of inertia and the final angular velocity, I also calculate the ring's final angular momentum, L. You can see that $$\bold{\omega}$$ and L are in the x_1x_2 plane.

I would think that the ring immediately starts to undergo torque-free precession about its constant angular momentum vector. But here's the spooky part: for a ring, the spin component of its angular velocity (the contribution from the spin about the ring's axis) makes an obtuse angle with the angular momentum. Immediately after impact, the ring's axis is still along the x_1 direction. So it seems the direction of the ring's spin flips 180 degrees right after impact. Does that seem weird?

 

[LightHero]

Is it physically possible? If so, how?The answer to this question is yes, it is possible for the ring's spin direction to flip 180 degrees immediately after impact. This is because the ring's angular momentum, L, is not necessarily aligned with its axis of rotation. When the ball collides with the ring, the angular momentum is transferred from the ball to the ring. Since the angular momentum of the ring is not necessarily aligned with its axis of rotation, the ring can experience a torque that causes the direction of its spin to flip.

 

[astrokreng]

I would first like to commend you for your thorough analysis and use of conservation laws to solve for the resulting motion of the ball and ring system. From your calculations, it does seem that the ring will undergo torque-free precession about its constant angular momentum vector after the collision.

Regarding the "spooky" part of your question, it is not uncommon for the direction of the spin to flip after a collision. This phenomenon is known as the "tennis racket effect" and has been observed in various systems, including spinning tops and tennis rackets. It occurs because the direction of the spin is not necessarily aligned with the angular momentum vector, and the collision can cause a change in the orientation of the spin axis.

While it may seem strange, this behavior is actually a consequence of the conservation of angular momentum. The spin component of the angular velocity is not necessarily aligned with the angular momentum vector, and the collision can cause a change in the orientation of the spin axis to conserve angular momentum.

Overall, I would say that your analysis and conclusions are sound, and the flipping of the spin direction after the collision is a natural consequence of the conservation of angular momentum.