Which ball spins the longest? I just thought of a mind-boggling question about angular velocity and friction.
Here is the picture:

The first figure shows a ball spinning around the y-axis, and the seconds shows another ball that rotates about the z-axis.
In which figure would the ball spin the longest until its angular velocity will be zero radians/second?
The force of friction and gravity, the mass and the radius of the balls, and the initial angular velocity for both scenarios are the same. Both balls are on a flat surface. Assume air resistance is negligible. Assume the ball maintains a solid shape at all times.
Many of my friends say that it's the second figure. My physics teacher and I say they will stop in the same time. Who has right?
 A: Note that the rolling ball has more kinetic energy than the spinning ball, because it has both angular and linear velocity, compared to just angular velocity. So, stopping the rolling ball requires more work than stopping the spinning ball. If the work per time done by friction is the same in both cases, it will take longer to stop the second ball. As it stands, I'm not 100% sure how to compare the friction forces involved in spinning vs. rolling, so if anyone can support their equivalency, it would improve this answer!
A: Let's focus on the second ball, the one rolling on the floor. If it rolls without slipping, then it will stop spinning when it comes  to rest, or when its total momentum is zero.
How quickly does its total momentum decrease? Newton's 2nd law tells us the rate of change of momentum of an object is given by the total external force on the object.
What are the horizontal external forces on the object? This is where you have to be careful. Only static friction comes into play because the ball doesn't slip as it rolls, and the force of static friction can be zero! In particular, if the ball is perfectly rigid and doesn't slip, the force of static friction between the floor and the point of contact will always be zero!
If this isn't clear, think about circular motion. The points on the edge of the sphere move in circles, so the net force on them must point towards the center of the sphere. Hence, the net force on the point of contact must be vertical, with no horizontal component from friction.
The only thing that can stop a perfectly rigid rolling ball is air resistance. The second ball will roll forever, while the first ball will slow down for the obvious reasons.
