Can a Sphere Roll Without Slipping on a Frictionless Surface?

[Saitama]

Homework Statement

A solid sphere of mass M and radius 1m rests on a smooth horizontal surface. A force F acts on sphere as shown in figure. If it rolls without slipping, find the value of H.

Homework Equations

The Attempt at a Solution

This problem was presented by the teacher today.

My question is, how it can start rolling without slipping in absence of friction?

The teacher solved the problem in the following way:

Torque about CM: FH=Iα or α=(5/2)(FH/M).
From Newton's second law: F=Ma or a=F/M.
From the condition of rolling without slipping a=α (radius is 1 m as per the question)

Hence, H=(2/5) metres.

But this doesn't make sense, I have read that friction is required to initiate rolling. In absence of friction, an impulse can be provided at a suitable height to start rolling but in the given question, a force is applied.

Any help is appreciated. Thanks!

 

[rcgldr]

As long as the force is applied, the sphere will experience linear and angular acceleration. With the proper value of H, the sphere will "roll" as it accelerates, meaning it's surface speed relative to the center of mass will equal the linear speed of the sphere.

 

[haruspex]

I have read that friction is required to initiate rolling.

That's a popular myth. As rcgldr says, rolling merely means that there is no relative motion of the surfaces in contact. Friction is the usual way for that state to be achieved, but it is not the only way.

 

[Saitama]

Thank you very much rcgldr and haruspex.