# Rolling and friction

I want to describe the motion of a ball that rolls without slipping in a plane where acts a force field not in the direction of the motion. To illustrate better, the ball is put on the origin, there is a field h in y direction, and a velocity v0 in the x direction. There is also a friction force parallel to the plane in a way that the balls always rolls without slipping in the plane. How can I calculate y in function of x?

http://imagizer.imageshack.us/v2/280x200q90/673/08e7ea.png [Broken]

I've tried to solve this, but I'm having problems. I know the only torque acting on the particle is Ff.R (Ff = friction force, R is the radius of the ball). So Ff.R = I.γ (I is inertial momment of the ball, γ is the angular acceleration). If I substitute I by 2/5 MR² and γ by a/R I get Ff = 2/5 Ma. But the problem is there, can I do this substitution? Also, is the direction of the friction force opposite to the instantaneous velocity of the particle? Is the ball is rolling in a way that $\vec{ω}$ is always perpendicular to $\vec{v}$? I don't know how to go on from there.

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## Answers and Replies

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rcgldr
Homework Helper
For the side force, you could imagine that the ball is rolling initially horizontally on an inclined plane, with force perpendicular to the initial velocity = m g sin(θ), where θ is the angle of the plane (from a level planel). The force is only initially perpendicular to the path; as soon as the ball starts to roll downwards on the inclined plane, a component of the force is in the same direction as the path of the ball.

Complicating matters is the fact that the ball's axis of rotation is yawing (relative to the plane), and I'm not sure if some type of gyroscopic precession effect would be involved and/or if a twisting torque occurs at the point of contact (assuming zero slippage).

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Delta2
Homework Helper
Gold Member
What sort of force field is h? Is it the gravitational field? if it isnt the gravitational field what is the formula for the force exerted by the h field to the ball? Do we also assume there is gravitational field in this problem (perpendicular to the x-y plane of the given figure)?

Delta2
Homework Helper
Gold Member
Well anyway seems to me that the condition imposed of "rolling without sliding" means that the velocity of the c.o.m is always equal and opposite of the linear velocity of the point of contact , hence the friction force is also in the opposite direction of the c.o.m velocity.

Well anyway seems to me that the condition imposed of "rolling without sliding" means that the velocity of the c.o.m is always equal and opposite of the linear velocity of the point of contact , hence the friction force is also in the opposite direction of the c.o.m velocity.
Sorry I forgot to say
There is a gravitacional field perpendicular to the Sheet, otherwise there would be no friction

Force exerced on ball is m h

See my article " The polygon model of rolling friction" in the journal IJEERT

This is not so simple. Rolling balls have angular momentum that change in direction by external forces:
$$\textbf{τ} = \frac{\partial\mathbf{L}}{\partial{t}}$$
and cyrcular radius varies.

As per Reynolds experiments the distance advanced by the wheel in on revolution is less than 2πR when the wheel is deformed at the contact area, which is in agreement with my model "Polygon Model of rolling Friction" The distance advanced by the wheel in one turn is only the perimeter of the polygon inscribed in a circle corresponds to the deformation.
Dr. Muhammed Arif M