Rolling Ball Mechanics: Proving Zero Horizontal Force Constraint

[Logarythmic]

Homework Statement

Consider a homogenous ball constrained to move on a horizontal plane.
Show that when the ball is rolling without slipping on the plane, then the total horizontal force on the ball must be zero.
HINT: Consider the ball as a rigid body and suitably combine the equations describing the rotation with the rolling constraints.

Homework Equations

None

The Attempt at a Solution

None. Been thinking about this for two days so I need a starter please.

 

[Kurdt]

Well a good place to start is with the hint and find out what the equations governing its rolling motion are and the conditions for rolling without slipping. So you can bet there's something to do with friction in there aswell.

 

[Logarythmic]

I'm not that stupid. ;)

 

[Kurdt]

What exactly are you stuck on then?

 

[OlderDan]

Homework Statement

Consider a homogenous ball constrained to move on a horizontal plane.
Show that when the ball is rolling without slipping on the plane, then the total horizontal force on the ball must be zero.
HINT: Consider the ball as a rigid body and suitably combine the equations describing the rotation with the rolling constraints.

Homework Equations

None

The Attempt at a Solution

None. Been thinking about this for two days so I need a starter please.

If you assume there is only a frictional force acting, then the statement is easily proven. Friction has to oppose the motion of the ball, so if there is any friction the ball will decelerate. However, if this is the only force acting the torque would produce an angular acceleration in the direction of angular motion, so the rotation rate would increase. This cannot happen if there is no slipping, in which case the rotation rate would have to decrease in proportion to the linear velocity.

A similar argument can be made about the application of a single horizontal force at the top of the ball. There is too much torque for the linear acceleration to keep up with the angular accelration. However, the torque can be reduced by lowering the applied force to a point below the top of the ball, so it is possible to match the angular and linear accelerations to have no slipping with a single horizontal force. The point of application to achieve this would be 2/5 of a radius above center. At lower points still above center, a frictional force can provide additonal torque while reducing the linear acceleration, and no slipping can be achieved with a net force acting and a net accelration.

I assume the problem is more restrictive than the application of the forces applied at arbitrary points, and is talking about a single constraining force like friction applied tangent to the ball.