Exploring the Physics of a Ball on an Infinite Plane

[brainpushups]

A person asked this question of me recently and it generated some discussion amongst the people in the room (many of whom had a limited background in physics). The original question went something like this: suppose that a ball is initially at rest in a wagon and the wagon is given a horizontal pull (so that the ball appears to roll to the back of the wagon). The wagon is then brought to an abrupt stop. Does the ball continue to roll backward relative to the wagon because that is the way the ball was spinning?

After some discussion the question was modified to be more like this:

Suppose that there is a ball which rests on an infinite horizontal plane. The plane is pulled horizontally so that it reaches a high speed with respect to its initial rest state and then the plane is brought abruptly to rest. What is the behavior of the ball a) as seen from the initial rest frame and b) as seen from the frame of the plane?

To avoid a rather lengthy first post I'll avoid writing my answer to the question for now, but I'm interested to see some responses. Kinda funny that this question is almost identical to Feynman's story of asking his father about the behavior of a ball in a wagon!

 

[A.T.]

The plane is pulled horizontally so that it reaches a high speed with respect to its initial rest state and then the plane is brought abruptly to rest.

Put a ball on a piece of paper, pull it out such that the ball starts rolling. What happens when the paper ends?

 

[brainpushups]

Have you tried this? It is hard to get consistent results. Also, the change in the friction coefficient would have an effect. If you try this you need to tape paper down to where the object will roll off.

 

[jbriggs444]

It is sometimes possible to make arguments from symmetry. This saves a lot of useless calculation.

Adopt a frame of reference in which the ball is initially at rest. The only external force on the ball is friction with the bed of the wagon. The distance from bottom of ball to its center of mass is constant. The moment of inertia of the ball is constant. The mass of the ball is constant. This means that linear acceleration is proportional to force which is proportional to torque which is proportional to rotational acceleration.

It follows that regardless of the whether the ball is slipping or not and regardless of the value of the coefficient of static or dynamic friction and regardless of how the frictional force varies over time the ratio of linear velocity to angular velocity of the ball is fixed. Further, the ratio is negative -- if the ball is moving to the left, it is spinning clockwise and vice versa.

If the wagon is at rest and the ball is not sliding, the only consistent state is with the ball is also at rest. [If the ball were moving to the left and spinning clockwise, it would be sliding and slowing down. Same if it were moving to the right and spinning counter-clockwise. If it is at rest linearly then it is also not rotating].

It does not matter what pattern of motion the wagon and ball went through before the wagon returns to rest. Once it stops sliding, the ball will always end at rest if the wagon is at rest.

 

[brainpushups]

That's a great explanation. My response was similar (though not as eloquent), but I did not include the case of skidding. The explanation was not well received. I think this is probably because the intention of the question was to imagine that the wagon comes to rest instantaneously.