Frictional force between two rotating cylinders

[Dayal Kumar]

Homework Statement

.A cylinder P of radius r_P is being rotated at a constant angular velocity ω_P along positive y-axis with the help of a motor about its axis that is fixed. Another cylinder Q of radius r_Q free to rotate about its axis that is also fixed is touched with and pressed on P making an angle θ between their axes. Soon after the cylinders are pressed against each other, a steady state ls reached and the cylinder Q acquires a constant angular velocity. What can you conclude about the direction of frictional force when the steady state is reached?

Homework Equations

The Attempt at a Solution

I am getting that cylinder q will be rotating with a fixed angular velocity in the steady state and also it's centre of mass will be translating with ω_Qr_Qtanθ along the fixed axis of rotation. I am unable to understand why there will be frictional force if there is no acceleration or relative motion.

 

[Chestermiller]

Have you provided the exact wording of the problem?

 

[BvU]

Hello Dayal,

it's centre of mass will be translating with ω_Qr_Qtanθ along the fixed axis of rotation

Would that be consistent with 'steady state' ?
And: would it be consistent with 'no frictional force in action' ?

 

[Dayal Kumar]

The wording of the question is correct.
I think I am not getting what exactly is happening at the contact point. In steady state there should not be any slipping between the surfaces, what will happen to the angular velocities of the cylinders in such a case, please explain??

 

[Chestermiller]

The wording of the question is correct.
I think I am not getting what exactly is happening at the contact point. In steady state there should not be any slipping between the surfaces, what will happen to the angular velocities of the cylinders in such a case, please explain??

I am not able to picture the geometry. Can you provide a diagram please.

 

[CWatters]

I believe it looks like crossed fingers.

I think there is relative motion between the two because of the angle.

Try drawing rings around both rollers representing the motion of the point of contact. Two points on these rings converge on the contact point at different angles.

 

[haruspex]

it's centre of mass will be translating with ωQrQtanθ along the fixed axis of rotation.

It says it is free to rotate about its axis. There is nothing about whether it is free to slide along its axis. I would assume not.

 

[Dayal Kumar]

I have now understood that there must be relative motion between the two and the steady state is reached when the direction of frictional force becomes parallel to the axis of Q such that it's angular velocity remains constant. Thank you for your instructive replies.

 

[haruspex]

steady state is reached when the direction of frictional force becomes parallel to the axis of Q

It is not very intuitive, but I agree with your answer. Thank you for posting such an interesting question.

 

[CWatters]

I'll have to think about it some more. I thought from symmetry the angle of the frictional force would be theta/2.

 

[Dayal Kumar]

There does not actually exist any symmetry because the motor attached to cylinder P makes it rotate at a specified constant velocity whereas the friction is responsible for making the cylinder Q reach to the constant angular velocity as required in the steady state.

 

[CWatters]

Ok that didn't take long. I agree with you. The frictional angle will change as Q accelerates but once the angular velocity becomes constant it stops parallel with the axis of Q so the isn't a tangential component.