My son and I were playing in the park and accidently came across an interesting physical force. We threw a marble on an angle through the inside of a plastic blow-mold cylinder. The cylinder was approx 1M long X 0.6M Diameter. The marble returned after reaching top dead center on the second loop. If thrown hard enough it would spiral from one end to the other until it dropped. We continued the excercise from each end at different speeds with the same result so long as the marble never left the surface of the cylinder. What causes the marble to return and how? Any ideas? Thanks!
I am having a hard time visualizing exactly what is going on. Is the cylinder axis exactly horizontal to the ground or slightly tilted? What do you mean by throwing the marble at an angle? The marbles motion in general can be broken into two components: the part along the direction cylinder's, and the part transverse to the axis. Along the axis, the marble is constrained only by friction so its motion in this direction is generally straight and constant, with a slight deceleration due to friction. Transverse to the axis, the cylinder exerts a constant inward force forcing the marble to trace out a circular orbit. The total motion is a circle in one plane plus a line along the axis, which gives you a spiral. At high speeds, gravity is negligible. At moderate speeds, gravity causes the marble to be slower at the top of its circle and faster at the bottom, like a pendulum. At low speeds, gravity overcomes the inertia of the marble which kept it pressed against the cylinder wall, and the marble falls out of its circular path. If the cylinder is perfectly horizontal, the marble will spiral away from you and not come back. If it is coming back to you, I suspect the cylinder is not horizontal and gravity is pulling it back.
It is a gyroscopic affect. Linear inertia would make it go through the cylinder along a helix, but that would mean that the axis of the spin (due to rolling) has to change (rotate around the cylinder axis). The result is a torque perpendicular to the current spin axis and the cylinder axis (and thus also perpendicular to the cylinder surface at current contact point). This torque makes the marble turn around, and come back. http://en.wikipedia.org/wiki/Gyroscope If the marble was sliding, not rolling, it would move on a helix along the cylinder, and not come back.
You are suggesting that this little glass marble massing about 20g has enough angular momentum stored that it will overcome all frictional forces to the contrary (as well as the comparatively huge kinetic energy imparted to it by the throw), and reverse its direction due to gyroscopic forces alone? No way.
So 20g are too little for angular momentum but enough for "huge kinetic energy" ? For a rolling object the net linear momentum and the angular momentum are proportional. The same applies to kinetic energy and angular kinetic energy. And what frictional forces does it need to overcome? It is rolling, and makes a clean turn, because the gyroscopic output torque is perpendicular to the surface.
Yes. You spin the marble as fast as you can. I will stop it spinning with the tip of my pinkie nail. My turn. I will throw it as hard as I can at you. You stop it with your front teeth.
We are not talking about the ability of humans to transfer different types of momentum to the marble with different body parts. We are talking about a rolling marble. For a rolling object the net linear momentum and the angular momentum are bound to each other.
I think you're talking about too ideal a case. The marble might be bouncing and skidding as much as it might be rolling. The OP did say it performed two loops, and it is a cylinder 60cm in diameter. It is moving way too fast for any gyroscopic motion to come into play.
On the contrary - ignoring the spin is idealsing too much. The marble reverses its linear momentum along the cylinder axis, and comes back. You know, the linear momentum that is so huge that it would knock out my teeth. If there is so much momentum transfer between the marble and the cylinder, there must be enough traction to make it spin so it mostly rolls. Well, that is easy to test: Throw something into a cylinder, that doesn't roll. A coin sliding on its flat surface for example. I expect it to do a helix.
I'm afraid I must withhold further speculation until I see a diagram or at least a better description. We don't know what he's experiencing or describing. We're all talking out of our hats. For all we know the cylinder is inclined at 45 degrees. Then we'd feel pretty silly trying to use gyroscopic motion to ratinoalize why the coin came back...
Here is a demonstration on this: http://demonstrations.wolfram.com/RollingBallInsideACylinder/ They consider a vertical cylinder, where without disspative forces the rolling ball thrown in from the top, would go up and down and never reach the bottom end. The same applies to a horizontal cylinder. The torque perpendicular to the surface at contact point, that makes the ball turn around is called "Coriolis torque". They also give this reference: http://ajp.aapt.org/resource/1/ajpias/v74/i6/p497_s1
But if you throw the ball from the top and it still rolls back to the upper end, then you gonna blame gravity for that?
But the constant of proportionality will be tiny in the case of a small sphere so there will be very little rotational energy compared with the translational energy. . A large diameter wheel, on the other hand . . . .
The ratio of rotational to translational kinetic energy for an object rolling straight does not depend on the radius. For a solid sphere it is 0.4 which is neither "tiny" nor "very little". But rolling straight is just the initial condition here. The ball soon gets a spin around the surface normal, so the ratio could get even higher. When you play around with the wolfram applet, you can get conditions where the KE due to that normal spin alone (red line) is greater than the entire remaining energy due to translation, roll rotation and gravity (blue line), in some phases of the loop: However, the below is closer to the situation with the marble in a big cylinder. The gravity is switched off, so it doesn't affect the motion along the cylinder axis, just like in a horizontal cylinder:
Are you telling us that the rotational energy is not a function of the moment of inertia? Obviously the ratio would be the same for all uniform spheres. That wasn't spelled out in the post, though.
No, I said: "The ratio of rotational to translational kinetic energy for an object rolling straight does not depend on the radius." If it's so obvious, then why do you point out that the sphere is small? You said: "But the constant of proportionality will be tiny in the case of a small sphere so there will be very little rotational energy compared with the translational energy" Yes, the 0.4 applies to uniform mass distribution. Since we talk about a marble I thought this was obvious. What kind of marble where you thinking about, where there is "very little rotational energy compared with the translational energy" during rolling?
If I got hold of the wrong end of the stick then so could someone else. It would be quite possible to imaging a marble which was a hollow sphere and then things would be different. (The ratio would be 1, I think) Or a sphere with a lot of mass at the centre, where the ratio could be as small as you like. Also there is an earlier post with a diagram of discs - which have a different MI from that of a sphere. The phrase 'uniform sphere' doesn't cost much to write and gives helpful precision. That's all but perhaps I was being too picky.
No, it would be 2/3. We talking about a marble here. Why would one assume such a non-uniform mass distribution? I said 'solid sphere', and from the context (marble) it was obvious that I meant uniform density. If you are so much into precision in language then you should have specified what mass distribution you assumed where the ratio is tiny, because it had obviously nothing to do with the marble discussed here. Instead you mentioned the size, which is irrelevant for the ratio.
Yes - 2/3. I was thinking of a circle - not a spherical shell - durr. Yes - I agree that size of similar objects has no bearing on it.