# A pure rolling sphere problem

Question: A sphere $$S$$ rolls without slipping, moving with a constant speed on a plank $$P$$. The firction between the upper surface of plank $$P$$ and the sphere is sufficient to prevent slipping. Friction between plank and ground is negligible. Initially plank $$P$$, is fixed to ground with help of a pin $$N$$. How will motion of sphere and plank be affected if, the pin $$N$$ is suddenly removed?

According to me the plank should move in opposite direction to that of sphere, since centre of mass must remain at rest.

But according to my teacher's explanation, the motion of plank and sphere will remain unaffected because since in question, it is given that that sphere is moving with constant speed, which means no acceleration. This implies there is no force acting hence, there would be no friction acting between plank and sphere. This explanation seems odd to me.

Please give appropriate explanation and point out any faults in any of the explanation mentioned above. Any suggestions are massively appreciated.

This is indeed a tricky concept. Consider a perfectly rigid cylinder on a perfectly rigid horizontal plane moving in a direction perpendicular to its axis. Let the radius of the cylinder be $$R$$, its velocity be $$v$$ and its angular velocity by $$\omega$$.

If there is no friction between the cylinder and plane, the cylinder will continue with constant $$v$$ and $$\omega$$ forever. It is not necessary for $$v=R\omega$$. $$v$$ and $$\omega$$ are independent of each other.

If there is sufficient friction between the cylinder and plane and $$v \neq R\omega$$, friction will act such that the cylinder ends up with $$v=R\omega$$. At this point, once $$v=R\omega$$, there will be no friction at all. The cylinder will continue with $$v=R\omega$$ forever. The velocity of the point of contact with respect to the plane is zero.

It's the same idea with a block on a rough table: If no force pushes on the block, and the block has no velocity relative to the table, there will be no friction.

So in summary, friction only does work when the point of contact of the cylinder "attempts" to slip with respect to the surface. In other words, when tackling problems like this, you must consider whether $$v=R\omega$$ will still be satisfied if you "switch off" the friction. If it is still satisfied even without friction, then friction does not act at all.

It might come as a surprise to you that the frictional force between the sphere and the horizontal plank when the sphere is rolling without slipping along the plank at constant velocity is zero.
This must be so as the only possible horizontal force on the sphere is the force of friction on it due to the plank and this horizontal frictional force on the sphere, if it existed, would mean that the sphere would not be moving at constant velocity.
So there is no horizontal force of interaction between the sphere and the plank.