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Main Question or Discussion Point
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 [tex]\bold{\omega_i}[/tex] 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 [tex]\bold{\omega}[/tex] 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 [tex]\bold{\omega}[/tex] and L are in the x_1x_2 plane.
I would think that the ring immediately starts to undergo torquefree 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?
Shown in the "before impact" diagram are the unit vectors of the coordinate system. The initial angular velocity [tex]\bold{\omega_i}[/tex] 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 [tex]\bold{\omega}[/tex] 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 [tex]\bold{\omega}[/tex] and L are in the x_1x_2 plane.
I would think that the ring immediately starts to undergo torquefree 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?
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