# How can angular velocity be constant even when there is a torque by friction?

A cylinder at rest lying on a rough ground is given an impulse which imparts a translational velocity (no angular velocity) to it. The question goes ahead with finding time after which rolling starts etc. etc.

But I have another question: The friction from the ground will decrease the translational velocity but increase the angular velocity (as the torque of friction will be in the direction of rolling). After some time, the translational velocity will become equal to angular velocity and hence pure rolling will start.

After this time work done by friction will be 0 as the point of contact is not moving and hence translational velocity will be constant. Now my doubt is that, when translational velocity is constant, it implies angular velocity will also be constant (as translational velocity is a multiple of angular velocity). But the torque from friction (about the centre of mass) will not be zero which means angular velocity will not be constant. In some book where there was the same problem it was written friction will vanish. How can friction vanish? How can anguluar velocity be constant even when there is a torque by friction?

• Is there drag on the purely rolling cylinder or not? If not, then it will roll forever and no work is done. If so, then work will be done by the drag (e.g. non-elastic compression of the rough ground). Which is it? – Daniel Griscom Aug 19 '15 at 0:12
• "How can friction vanish?" - Friction disappears when there is no relative motion between the cylinder and the surface. So there is no torque being applied by friction. If the cylinder were to suddenly hit an ice patch, it would continue to roll exactly the same as it was before. – mbeckish Dec 7 '15 at 14:30

## 3 Answers

In theory there is no torque from friction once the cylinder is rolling. Imagine that there is no supporting surface but no gravity - then the cylinder will just continue to "roll" forever. So all the supporting surface needs to provide to the cylinder is a force through its centre of mass to counteract the gravitational force. This is a force perpendicular to the cylinder surface at the point of contact. It is not a frictional force. There is no torque necessary.

I hope that helps.

If there is a friction force (static or kinetic) then there will also be a torque. Both the linear and rotational velocities of the cylinder will be either accelerating or decelerating depending on the direction in which the friction is acting.

When this acceleration/deceleration has finished the cylinder will have reached an angular speed at which there is not only no slipping at the contact point but also no tendency towards slipping. There is then no longer any friction force because there is no longer any tendency for the two surfaces to move relative to each other.

Inertia keeps the cylinder rotating without any need for friction, just as it would if the surfaces were perfectly frictionless.

In practice there is also some rolling resistance - sometimes also called friction - which slows the cylinder. This arises from the deformation of the surfaces and hysteresis - energy stored in the deformation is not perfectly returned after the load is removed. This resistance can be very low when the surfaces are very hard (eg glass marble rolling on steel plate).

Once the cylinder is rolling, the velocity of the point of contact of the cylinder with the ground is zero. Since there is no relative motion between the cylinder and the ground, the force due to friction, and hence the torque due to friction will be zero.