## rolling without slipping - too many equations

1. The problem statement, all variables and given/known data
I think I have too many equations for unknowns for rolling without slipping. Suppose a wheel is pushed and left rolling along the ground without slipping.

2. Relevant equations
T = I*alpha, where alpha is the rotation about the wheel's centroid and I is the moment of inertia.

F = ma.

omega = r*v, where omega is the rotational speed, r is the wheel radius and v is the translational speed. The rolling without slipping condition. From this, alpha = r*a.

T = r*F, where F is friction.

3. The attempt at a solution

I*alpha = r*F, so alpha = r*F/I. For rolling without slipping, a = r^2 * F/I. But from Newton's second law, a = F/m. What am I doing wrong. Thanks.
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 There are 3 'F' here not one. F force apllied Fnet=ma Ffriction=Iα Fnet=F apllied±Ffriction=ma

 Quote by azizlwl There are 3 'F' here not one. F force apllied Fnet=ma Ffriction=Iα Fnet=F apllied±Ffriction=ma
What is the force applied you're referring to? I was referring to the situation after the person/external agent releases the wheel with some initial velocity, and the wheel is just rolling by itself on the ground without slipping subject to friction.

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## rolling without slipping - too many equations

So the wheel is just rolling along with a = 0, since there's no longer a force pushing it?
 Without external forces, the wheel will keep on rolling forever at initial rotational and translational velocity.
 What, according to you, would happen if someone rotates the wheel clockwise and A) there is no friction B) there is friction. In this case what will be direction of friction?
 The constraint is rolling without slipping. Then friction as the torque for rotation. A. Will rotate forever like the satellite. B. The friction is anticlockwise.

 Quote by Doc Al So the wheel is just rolling along with a = 0, since there's no longer a force pushing it?
right, no one is pushing it, but there is friction. sorry if I was unclear in setting up the problem. Say I take a wheel, cylinder, ball or whatever and set it rolling on the ground, as you might pitch a bowling ball. It has a certain initial velocity but I am no longer pushing it. Friction is still present, but it is not dissipative, so there is rolling without sliding.

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 Quote by jason.farnon right, no one is pushing it, but there is friction. sorry if I was unclear in setting up the problem. Say I take a wheel, cylinder, ball or whatever and set it rolling on the ground, as you might pitch a bowling ball. It has a certain initial velocity but I am no longer pushing it. Friction is still present, but it is not dissipative, so there is rolling without sliding.
Assuming you have met the conditions for rolling without slipping on a horizontal surface, the friction will be zero.

 Quote by Doc Al Assuming you have met the conditions for rolling without slipping on a horizontal surface, the friction will be zero.
I don't follow. If friction were zero it seems to me there would be no rolling at all. The object would just slide.

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 Quote by jason.farnon I don't follow. If friction were zero it seems to me there would be no rolling at all. The object would just slide.
If friction were nonzero, then it couldn't be rolling without slipping at some constant speed. Friction is needed to change the rotational speed, but the friction force becomes zero once the conditions are met for rolling without slipping.
 You have to be careful and distiguish between 2 types of friction....sliding friction and rolling friction. When something like a snooker ball is first struck the ball slides over the surface and sliding friction causes the balls linear speed to decrease. The frictional force acting on the ball produces a turning effect which makes the ball rotate. At the point of contact with the table the forward velocity will decrease to a value, v, and the (backwards) rotation velocity will increase to a velocity, v. At this point there is no relative motion between the point of contact on the ball and the point of contact on the table so sliding friction will become zero. The ball will then continue to roll with no sliding (slipping). The link between linear velocity and angular velocity when this happens is v = ωr There is now only rolling friction which is very small compared to sliding friction

 Quote by truesearch You have to be careful and distiguish between 2 types of friction....sliding friction and rolling friction. When something like a snooker ball is first struck the ball slides over the surface and sliding friction causes the balls linear speed to decrease. The frictional force acting on the ball produces a turning effect which makes the ball rotate. At the point of contact with the table the forward velocity will decrease to a value, v, and the (backwards) rotation velocity will increase to a velocity, v. At this point there is no relative motion between the point of contact on the ball and the point of contact on the table so sliding friction will become zero. The ball will then continue to roll with no sliding (slipping). The link between linear velocity and angular velocity when this happens is v = ωr There is now only rolling friction which is very small compared to sliding friction
Thanks, that is something I was wondering about. But, maybe I'm missing something, but I don't see that it answers my original question. Take the point that there is only rolling friction and thereafter. I don't see what is wrong with my two derivations for the acceleration of the ball, but they are contradictory.

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 Quote by jason.farnon But, maybe I'm missing something, but I don't see that it answers my original question. Take the point that there is only rolling friction and thereafter. I don't see what is wrong with my two derivations for the acceleration of the ball, but they are contradictory.
Once the ball is rolling without slipping, there is no acceleration. (Ignoring rolling friction.)

Your derivation takes "F" as some force acting on the wheel. What's that force? The applied force? But that's zero, since you stopped pushing. Friction? There is none. (I'm talking static friction, not rolling friction.)

 Quote by Doc Al Once the ball is rolling without slipping, there is no acceleration. (Ignoring rolling friction.) Your derivation takes "F" as some force acting on the wheel. What's that force? The applied force? But that's zero, since you stopped pushing. Friction? There is none. (I'm talking static friction, not rolling friction.)
I was using F to refer to the "rolling friction". Why does that affect the derivation? The translational effect on the centroid of the wheel/ball/object should be analyzable separately from the spinning wheel, shouldn't it? I have encountered derivations where that appears to be the case.
 Rolling friction is static friction. Just like a block on the a rough table, the static fraction is equal or less to the force applied. When no force applied, static friction is equal to zero. When no force applied on rotating object without slippage then slipping friction is zero.

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 Quote by jason.farnon I was using F to refer to the "rolling friction".
I don't think you are. The term "rolling friction" usually refers to a dissipative force caused by the deformation of the contact surfaces. When truesearch used the term I think he actually meant the static friction that appears during rolling motion. That static friction force is zero, once rolling without slipping is attained.

Actual rolling friction is a bit more complicated and is the force that will eventually stop the wheel as it rolls along the horizontal surface.