Rolling without slipping and torque

[aaaa202]

Just solved a problem where I found the particular coefficient of friction that a ramp would need for a body with a certain moment of intertia to roll down it.
So let's say i determined this to be 0.5. If now i try to roll my body down a surface with a lower coefficient it wouldn't be able to roll since there wouldn't be enough torque to produce a sufficient angular velocity?
But what if it was bigger? Would it then spin too much?

 

[Ken G]

The amount of spin (without slipping) is determined only by the rolling speed, not by the coefficient of friction. So changing the coefficient of static friction doesn't change anything about the rolling, so long as it actually is able to produce rolling. Increasing the coefficient makes it even better at maintaining rolling, so increasing it won't do anything to a motion that is already rolling, but reducing it could cause slipping as you say.

 

[aaaa202]

hmm I don't see it. Please elaborate on that, because as I see it a greater friction would have the potential to exert a bigger torque and thereby increase the angular velocity.

 

[maCrobo]

you have to think that the coefficient of friction gives you the maximum amount the friction force can take. Indeed, F=µN is the maximum friction force applicable on a body.

Think about that: a body on a horizontal plane with a friction coefficient µ. If you apply a force F>µN the body will start moving; if you apply a force less than µN the body maintain its position, so the resultant of forces on it is zero, it means that even the friction force must be less than µN in order to balance the force applied (that doesn't make the body move).

The same is for rolling, if the coefficient is greater it just means that your body has more grip on the surface and will roll with a great angular velocity as well as a small one, because the friction force will fit on it.

 

[WaterAndSand]

Indeed. Let's say this object is a uniform cylinder, once there is sufficient friction to prevent said cylinder from slipping as it rolls, increasing the friction coefficient isn't going to make it roll any faster, only better at rolling at best. The surface isn't providing any extra force, gravity remains 9.8m/s. If the coefficient is reduced, the cylinder will still roll, but will slip dependent on the coefficient.

 

[Doc Al]

Just solved a problem where I found the particular coefficient of friction that a ramp would need for a body with a certain moment of intertia to roll down it.

How did you find it? I suspect that you found the minimum coefficient of friction to support rolling without slipping.

 

[Ken G]

hmm I don't see it. Please elaborate on that, because as I see it a greater friction would have the potential to exert a bigger torque and thereby increase the angular velocity.

Maybe what isn't clear to you is that a rolling object always has a definite relationship between the speed of its center of mass, and the speed of its spin. That has to hold, no matter what the coefficient of friction is, if it is rolling. Since conservation of energy on the incline can tell you the total of the kinetic energy of the center of mass motion, plus the kinetic energy of the spinning motion, and the connection between the two is completely given by the shape of the object, you always know exactly how fast it is rolling at any height-- so long as it is rolling. All this before you know anything about the coefficient of friction! But then, you ask, is the coefficient of friction really going to allow rolling? That's when you need to do the calculation you did, to get the minimum coefficient that would allow rolling. Any larger coefficient would then not change the motion at all. As mentioned above, the coefficient of static friction does not give the frictional force, it gives the maximum frictional force. The actual force is given by whatever is required to achieve rolling.