# Does a ball that rolls inside a rotating wheel experience centrifugal force?

Can you help me with a 'contention' I have with my university supervisor? Apologies if this has been answered elsewhere.

I have a device that can be modeled as a wheel/tire with a ball inside (in 2D) - a vertical circular track with a ball which is able to roll freely inside. At rest, the ball is at the bottom of the vertical circle and that will be the angular position of the ball about the center of the wheel.

I want to understand the governing equations for the motion of the ball if the wheel is rotated.

My understanding is that if the wheel is rotated at a certain angular velocity, the ball will be dragged up one side until gravity pulls the ball back towards the zero position. The torque from the friction will cause the ball to roll. At a certain point it will achieve equilibrium, the ball will roll constantly towards the bottom of the circle but not move, maintaining an angular position ($$\theta$$) somewhere between 0 and 90 degrees - a bit like a hamster running in its wheel!

My question is if the ball is rolling fast enough to maintain its position inside a rotating track, does it experience a centrifugal force? Everything else being equal, would 2 balls of different masses roll at different theta positions? Is it possible to predict this value of theta for different balls?

Thanks for any help!

• I don't believe it makes any sense to talk about centrifugal force, unless you are trying to explain the motion of the ball in a coordinate frame that is rigidly attached to the rotating tire. I'm not sure I would even know where to begin to write equations for that. It's much easier in the non-rotating lab coordinate system because, in that system, according to your description, the ball isn't even moving (except for it spinning about its own center, of course.) Jun 12 '20 at 12:47
• whether or not centrifugal force applies in this seems to depend on how you look at it. which confuses me! the actual device involves coils and magnets and i need to model it in matlab, I have been advised to simplify it by flattening out the circle and model it as if the track is stationary and the ball is moving. so I think in tht case i need to use a frame of reference from a staionary point on the circular track. so if the track is stationary and the ball is rolling around in a circle, it is accelerating towards the centre and this needs to be balanced by the centrifugal force? Jun 12 '20 at 13:09
• Re, "...seems to depend on how you look at it." Yes. Centrifugal force is a fictitious force (a.k.a., "pseduo-force") that only exists in rotating coordinate systems. I know that for a stationary object in a rotating coordinate frame, the force is directed radially outward from the origin, and its magnitude is proportional to the distance from the origin; but I'm not competent to explain the force felt by an object that moves in a rotating coordinate frame. Jun 12 '20 at 13:29
• Do you mean to ask about centripetal force that does not depend on the reference frame? Jun 12 '20 at 13:43
• Re, "...coils and magnets...flattening out the circle..." If some person told you that it was a good idea to model your apparatus (which you have not fully described to us) in a particular way, then perhaps that person is best qualified to explain why they thought it would be helpful to model it in that way. Jun 12 '20 at 13:52

Once the ball comes up to speed, the equilibrium position is at the bottom of the wheel (ignoring friction with the air). If it were up the incline, static friction would be causing an angular acceleration. The size of the ball does not matter. In this position, the ball is not translating or moving in a circle. It experiences no centripetal acceleration or centripetal force. The normal force supports the weight.

If an object is stationary, or in an inertial frame -as your rolling balls are- they do not experience centrifugal force. Imagine that the ball is not quite touching the wheel, is in very nearly the same equilibrium position, and is spinning the same as the ball in your scenario. It will behave as if there were no wheel; and it will obviously feel no centrifugal or centripetal force. Bring the ball into contact with the wheel and there will be no change. The presence of friction complicates the analysis a bit, but the underlying principle is the same: if an object is not following a curved or otherwise accelerated path, it is not going to experience centrifugal/centripetal forces.

• If an object is stationary, or in an inertial frame -as your rolling balls are- they do not experience centrifugal force. I would argue that objects never experience centrifugal forces. They are more of a way to make the math work out with Newton's second law for rotating frames. Stationary objects in one frame can be observed to have centrifugal forces acting on them in rotating frames. Jun 12 '20 at 13:58

Centrifugal force depends entirely on the chosen reference frame. It's consequences do not depend on the mass of an object (which cancels out from any equations). If you choose a reference frame in which the ball is stationary, there will be a centrifugal force, balanced by a normal force from the cylinder, and you only need this to be enough to generate a friction force sufficient to counteract gravity.

You appear to be doing variant of the "wall of death" which you can google. Once sufficient speed is obtained (as viewed from a non-rotating reference frame) it is easy to maintain a constant height, in which a vertical component of friction balances the force of gravity. Increasing velocity beyond this point does not change the analysis, and nor does mass.