Ball revolving inside of a cylinder.. Physics..?

Question

A hallow cylinder is positioned vertically such that its axis is orthagonal to the ground. A small ball is placed inside of the cylinder and made and made to revolve around the inside of the cylinder. Picture a marble whirling around the inside of a coffee can that is sitting in a normal position. The coefficient of static friction between the ball and the cylinder wall is assumed to be infinite, in other words the points of contact of the cylinder and ball have itentical velocities and no slipping occurs provided there is any arbitrary amount of normal force. The question: at what speed, if any, will the ball continue to revolve around the cylinder without spiraling down into it? Keep in mind that the ball has angular momentum parallel to its axis of rotation, and that the ball cannot slip down the sides, it must roll. Keep also in mind that if the angle of the balls axis changes to that of the cylinder then the circle defining the points of contact will no longer be the equator.

Ahh yes, zero area of contact, that does draw a distinction between motorcycles and idealized spheres, but fails to explain why it necessarily spiral down. Pointing out a difference between an a successful method and an unproven method does not disprove it. I wish I might allude to the equation so conveniently produced by zee_prime, but I'm not sure of the unexplained logic behind it.

Yess, oops, the last sentence is supposed to say changes FROM no TO

Answer 1

I maintain that at any finite speed the ball will spiral down. The angle the spiral makes with the horizontal will be equal to arctan gr/(v^2) where r=radius of the can, v is the velocity of the ball and g is acceleration due to gravity. The reason a motorcyclist can keep circling inside a wall of death indefinitely is that the bike wheels are elastic cylinders, not hard spheres, and they have a finite area of contact with the wall. A hard sphere has zero area of contact.

Answer 2

Unless I'm completely misunderstanding the question, this is simply not possible. If the cylinder's walls are vertical it is not possible.

A normal force acts on the ball, which causes the circular motion around the cylinder.
The weight W acts through the center of the ball. The only thing that can possibly oppose that weight, is an upward frictional force, f which acts on the side of the ball. Because these forces are not colinear, they result in an unbalanced moment which will cause the ball to roll down the cylinder.
There is simply no way to prevent this unless the cylinder wall is inclined.

*EDIT*
To augment zeeprime's response, a motorcyclist stays up because he can angle the motorcycle in such a way that a component of the normal reaction counteracts the effect of gravity. The spherical ball cannot do that.
This is not a difficult problem. It's basic dynamics.

Answer 3

Julie, you basically desire 2 formulae to remedy this project: If the radius = r, quantity of a cylinder = top x (area of the top) = h x Pi x r^2 quantity of a sphere = 4/3 x Pi x r^3 do no longer forget approximately the diameter for this reason is 4 cm, so the radius is 2cm. Now calculate the quantity of the cylinder, and subtract the mixed quantity of the six balls, and the end result's the respond. in case you do no longer understand why this must be, draw a image of the six balls all merely installation properly interior the cylinder and all will at as quickly as become sparkling.

Answer 4

I just ran into this problem in helping a student with her analysis of uniform circular motion equations. The concept understanding required looking at the coefficient of friction for static, rolling, and kinetic. We are stating that the ball and surface have friction opposing the velocity of the ball on the surface. We are looking for the ideal where the vertical forces on the ball at the surface are zero. This can only be true if the weight of the ball equals the force that results from a resistance to roll down the surface. We define that force as proportional to the surface force of the cylinder. One can look in tables and find that for a steel marble on a steel surface that the frictional force/ surface force is 0.6 for a rolling sphere on a steel surface. Then the linear velocity of this ball is given by w x r (radius of cylinder, cycles per second around the cylinder). We have acceleration equals v^2/r for uniform circular motion. We have the mass of the steel marble. We can then calculate the surface force on the ball. We also have the weight of the ball. Then if the frictional force that can be generated is greater than or equal to the weight, the ball will not spiral down. Using a coefficient equal to that found in the table, the required linear velocity of the marble can be calculated. Any value greater than or equal to this will result in no spiraling down of the ball. Once the linear velocity decreases below this threshold the ball will spiral down. This is all just back of envelope, first blush analysis. as much more physics, e.g. inertial top pertains to this problem....but at this level I believe the question is just asking for an understanding of uniform circular motion force diagrams and friction as presented at this level.

Answer 5

This is a pretty hard problem! I recall solving such problems about 50 years ago, but it would take too long to recall everything needed for me to attempt to solve it now.

If memory serves me correctly, however, the GENERAL solution involves (perhaps surprisingly) motion trapped between two horizontal layers. As is usually the case for such problems, there is a CRITICAL value for the speed at which the ball will simply roll around without slipping, at a constant height. THAT critical speed is presumably what you are looking for.

Best of luck to anyone attempting this --- it's a problem worthy of the Cambridge Mathematics Tripos!

Live long and prosper.

P.S. Did you mean "... changes FROM that of the cylinder..." rather than changes TO that of the cylinder..." in your final sentence?