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I just thought of a mind-boggling question about angular velocity and friction.

Here is the picture:

enter image description here

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?

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  • $\begingroup$ @Nathan Ramanathan 'Let the debate begin' Not a place to debate. $\endgroup$ – Wrichik Basu May 9 '17 at 20:22
  • $\begingroup$ That's weird for some reason the picture works in the computer but not in the phone. I added a link. $\endgroup$ – Nathan Ramanathan May 9 '17 at 20:32
  • $\begingroup$ This has everything to do with the friction involved. There is a different amount of friction given the two different spins regardless of coefficient of friction being the same. Unless you are saying there is zero(0) friction at all. In that case they are equal. $\endgroup$ – Brad S May 9 '17 at 20:43
  • $\begingroup$ But aren't both spheres touching the ground at one point at every second. In physics class they said that friction doesn't care about surface area, instead it depends on the normal force and the coefficient of friction, in this case kinetic friction. Therefore, the friction is the same in both scenarios. $\endgroup$ – Nathan Ramanathan May 9 '17 at 20:51
  • $\begingroup$ The availabality of the picture probably depends on that you have an active google drive session or not. I solved the problem (by inserting the picture here). $\endgroup$ – user259412 May 9 '17 at 20:55
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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!

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  • $\begingroup$ More work! Wouldn't that break the law of conservation of energy. According to kinetic rotational energy, which equals 0.5(I)(w^2) which is equivalent to 0.5mv^2) Both balls have the same moment of inertia and angular velocity therefore both scenarios require the same amount of work to stop. $\endgroup$ – Nathan Ramanathan May 9 '17 at 23:19
  • $\begingroup$ Therefore, the total kinetic energy for the second ball splits up into both rotational and translation kinetic energy which equals the total kinetic energy of the first ball. $\endgroup$ – Nathan Ramanathan May 9 '17 at 23:27
  • $\begingroup$ @NathanRamanathan Both balls have the same mass and same angular velocity, hence the same rotational kinetic energy. The second ball ALSO has translational kinetic energy, which the first ball does not have. The total kinetic energy of the second ball is indeed split into rotational and translational kinetic energy, but that total is not equal to the total kinetic energy of the first ball. You seem to be hung up on the idea that both balls have the same total kinetic energy, which simply isn't the case. $\endgroup$ – Nuclear Wang May 10 '17 at 12:42
  • $\begingroup$ May you explain how pushing a ball a certain way gives it a different amount of total mechanical energy? The same amount of torque is exerted upon both balls. Don't they have the same initial velocity? I am still confused about which ball will move the longest. $\endgroup$ – Nathan Ramanathan May 10 '17 at 19:58
  • $\begingroup$ The balls are not pushed the same way in both cases, otherwise they would be spinning the same way. The question makes no mention whatsoever of torque, it just states that they have the same initial angular velocity (but not the same linear velocity). Image both balls suspended by a string on a wheeled apparatus. Getting each ball to spin requires the same energy. For the second ball, you also have to push the whole apparatus across the floor, requiring more energy. $\endgroup$ – Nuclear Wang May 10 '17 at 20:13
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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.

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  • $\begingroup$ There is no stipulation that the ball is perfectly rigid. It appears to be a completely real-world scenario, meaning the second ball will not roll forever. If we abstract those real-world forces away, you could say the first ball spins forever, too. $\endgroup$ – Nuclear Wang May 9 '17 at 21:05
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    $\begingroup$ Nope, the reasons are not obvious at all ;) a ball that can roll without friction is so perfectly spherical, that it will touch the floor in exactly one point. So what is the torque in the first case? ;) $\endgroup$ – Ilja May 9 '17 at 21:06
  • $\begingroup$ Good points. Any assumptions that make one case trivial also make the other case trivial. This is a more complicated question that it seems. My instincts still say that for the same material, contact patch area, etc. the rolling resistance will be a smaller effect than the twisting resistance, but I can't prove it. $\endgroup$ – Luke Pritchett May 10 '17 at 1:24

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