Rolling with slipping and conditions for not slipping

[cromata]

Suppose that we have a some rotating object (lets say a wheel with radius R). Let's observe this problem from some reference frame in which center of mass translates with some velocity v and rotates with angular velocity ω. I know that condition for rolling without slipping is v=ωR (point at which wheel touches ground doesn't move). Also, I know that is only possible if coefficient of static friction is large enough so that point that touches ground has velocity 0.
But I don't really understand how to determine if the friction is large enough to cause rolling without slipping: how does it depend on the angular velocity of the wheel (or some other parameters)?

 

[andrewkirk]

Once the object is rolling, and obeying the equation you gave, the friction becomes irrelevant. In an ideal frictionless setup it would continue moving as if it were rolling even if the surface became frictionless. Imagine a flywheel rolling along a carpet and then going off the end of the carpet onto ice. Conservation of momentum and angular momentum would dictate that nothing changes.

Where friction is important is when there is angular acceleration. This is viscerally plain when a vehicle skids when too much acceleration is applied (think drag cars doing burnouts) or it brakes too hard. The angular velocity of the wheel plays no part in the calculation.

 

[drvrm]

But I don't really understand how to determine if the friction is large enough to cause rolling without slipping: how does it depend on the angular velocity of the wheel (or some other parameters)?

some simple experiments can be designed to study and understand the stages of rolling and the limiting conditions for rolling.

the following study may help you to see the role of rolling friction and it can be measured using a theoretical analysis as well-

ref.- https://billiards.colostate.edu/physics/Domenech_AJP_87%20article.pdf

 

[A.T.]

But I don't really understand how to determine if the friction is large enough to cause rolling without slipping: how does it depend on the angular velocity of the wheel (or some other parameters)?

You have to know all other forces and moments acting on the wheel. Then you can combine the force and moment equations via v=ωR (or its time derivative a=αR), and solve for the required frictional force. If its magnitude is less than the normal force time static friction coefficient, it will roll. Otherwise it will slide, so you use dynamic friction coefficient to get the friction force and work out a and α from that.

 

[cromata]

You have to know all other forces and moments acting on the wheel. Then you can combine the force and moment equations via v=ωR (or its time derivative a=αR), and solve for the required frictional force. If its magnitude is less than the normal force time static friction coefficient, it will roll. Otherwise it will slide, so you use dynamic friction coefficient to get the friction force and work out a and α from that.

Let's assume that there is some external force creating torque on the wheel, but it doesn`t affect translation (it`s possible if the force is in vertical direction in our wheel example). If coefficient of static friction is not large enough for rolling without slipping, will then wheel translate with constant acceleration N*k/m? (where k is coefficient of friction)?

 

[PeroK]

Let's assume that there is some external force creating torque on the wheel, but it doesn`t affect translation (it`s possible if the force is in vertical direction in our wheel example). If coefficient of static friction is not large enough for rolling without slipping, will then wheel translate with constant acceleration N*k/m? (where k is coefficient of friction)?

As has already been pointed out, friction is only necessary for acceleration. Once the wheel is rotating, no friction is required for it to continue rotating. Moreover, there is no minimum static friction needed. Although, the less friction you have, the less acceleration you can get.

 

[A.T.]

Let's assume that there is some external force creating torque on the wheel, but it doesn`t affect translation (it`s possible if the force is in vertical direction in our wheel example). If coefficient of static friction is not large enough for rolling without slipping, will then wheel translate with constant acceleration N*k/m? (where k is coefficient of friction)?

Yes, assuming k is the coefficient of dynamic friction and no other horizontal forces are acting.

 

[cromata]

Thank you for your answers