# Friction and rolling

I've got 2 questions which I tried asking my teachers and searching online , but none helped much. 1.) What role does friction exactly play in rolling? I know that without friction , the body would just rotate at its place and it prevents slipping, but what exactly does friction do? Does it provide the tangential acceleration to the lowermost point and if yes , what exactly does that do. 2.)About which axis does the body rotate. If It does about the centre of the body , the body will have a torque (provided by friction) which'll increase the angular velocity , thus resulting in the body not undergoing pure rolling?

• Could you describe the setup, it's unclear what is rotating and under what circumstances at present. Commented Dec 24, 2022 at 20:39

Question 1.

The friction force is opposed to the relative velocity between the object and the support of their contact points, applied at the contact point. This has the effect of decreasing the relative velocity until it vanishes. This means you’d have rolling without slipping occurs.

In the case of rigid bodies, you have two effects. Since it’s a force, it accelerates the motion center of mass. Also, since it is usually off center from the center of mass, it acts as a torque about the COM, it accelerates the angular velocity. It is beast to break down the problem in this order and then look at the effect on the contact point rather than the opposite, since you do not know what is the internal force (which preserves the rigidity of the object) that is applied on the contact point.

Btw, friction can also act directly as a pure torque (no line of action). It would be applied about the contact point, which opposes the angular velocity of the object with respect to the support. This has the effect of reducing relative angular velocity until it vanishes. This means you’d have pure slipping.

Question 2.

I’m assuming you’re asking from the frame of the support in the case of pure rolling. It’s counter intuitive, but the instantaneous axis of rotation passes by the contact point due to the non slip condition. You can check this on a simple example.

Hope this helps.

Assume a sphere of radius $$r$$ rotating at angular velocity $$\omega$$ with its centre of mass moving with a linear velocity $$v_{\rm c}$$.

If there is no slipping between the sphere and the ground, ie $$v_{\rm c}=r\,\omega$$, no frictional force is necessary to maintain the motion of the sphere.

Suppose that $$v_{\rm c}>r\,\omega$$ then to reach the no slipping condition the linear velocity of the sphere must decrease whilst at the same time the angular velocity of the sphere must increase.
This is achieved by a frictional force on the sphere in the opposite direction to the motion of the centre of mass of the sphere.

If $$v_{\rm c} then the frictional force on the sphere must be in the same direction as the motion of the centre of mass of the sphere.

About which axis does the body rotate?
If the body is rigid the angular velocity is the same about the centre of mass, the point of contact with the ground, and any other axis.
Consider three points on a disc or it could be a more general shape, $$A$$ at the centre, $$B$$ at the rim, and $$C$$ somewhere on the disc.

Relative to the xy-axes the disc rotates by an angle $$\theta_{\rm A}$$ clockwise about point $$A$$.
$$B$$ has moved to position $$B'$$ but note that relative to $$B'$$ and the xy-axes the line $$B'C'$$ has rotated by $$\theta_{\rm B} = \theta_{\rm A}$$ clockwise.
$$C$$ has moved to position $$C'$$ but note that relative to $$C'$$ and the xy-axes the line $$C'B'$$ has rotated by $$\theta_{\rm C} = \theta_{\rm A}$$ clockwise.