# Can rolling without slipping occur without friction?

Physics Asked by Laboni Gautam on August 31, 2020

If a body is rolling without slipping is it necessary that there is friction acting on it ? I encountered a question in which there is a spherical body and a force is being applied on its top point …so if there is only force then it should do translation motion only .. If there is friction also then it then only it can rotate with translation am I right?

As outlined in my answer here this is definitely possible.

If you have an object with moment of inertia $$I=gamma mR^2$$ on a horizontal surface and you apply a constant horizontal force $$F$$ some distance $$beta R$$ above the center of the object, then for rolling without slipping to occur we need a friction force

$$f=frac{beta-gamma}{1+gamma}F$$

So whenever $$beta=gamma$$ we would have rolling without slipping with no friction. Intuitively, in this case the accelerations associated with the forces and torques balance exactly in such a way so that $$v=Romega$$ without friction needing to alter translation and/or rotation to make this true.

Some examples (some might be hard to realize in reality, but work fine mathematically):

1. The force applied to the top of a circular hoop

2. The force applied a distance $$R/2$$ above the center of a cylindrical disk

3. The force applied a distance $$2R/5$$ above the center of a solid sphere

In your case with the force being applied at the top of the sphere ($$beta=1$$), there would need to be friction though.

Answered by BioPhysicist on August 31, 2020

Rolling without slipping occurs when for the revolutionary body (RB, i.e. sphere, cylinder, disc, ring etc) the following relation holds:

$$v=Romega$$ where:

• $$v$$ is the translational velocity,
• $$omega$$ is the angular velocity,
• $$R$$ is the object's radius.

Now imagine we make a sphere rotate at $$omega$$ and we then lower it carefully on a frictionless surface, so that the translational velocity vector is parallel to the surface and perpendicular to $$vec{omega}$$ and translational velocity scalar is $$v=Romega$$.

With no forces or torques acting on the sphere because the surface is infinitely smooth, the relationship $$v=Romega$$ holds forever!

Of course one may question whether such a motion really constitutes rolling without slipping. It looks more like slipping without rolling.

But if $$v or $$v>Romega$$ then only friction can 'correct' this until $$v=Romega$$.

Answered by Gert on August 31, 2020

If there is no frictional force between an object and the surface it is moving on, then there will be no relationship (or connection) between the rate (or direction) of rotation and the translational velocity. If a horizontal force is applied to the top of a sphere on a friction-less surface, it will cause (independently) both translation and rotation. If there is friction, then the two motions will be related.

Answered by R.W. Bird on August 31, 2020