@JohnRennie hi

Are u free

@Aladdin hi :-)

Yes I'm free

@DeepamSarmah hi :-)

@DeepamSarmah We can have pure rolling without any resistive forces like friction. It all depends on the linear velocity and the angular velocity imparted.

An object rolling at constant speed has a constant angular velocity i.e. the angular acceleration $\alpha = 0$. Yes?

On a frictionless floor, you can choose $v$ and $\omega$ wisely in a way $v=r\omega$. Then you'll have pure rolling.

For an advanced answer, I'll leave it to @JohnRennie sir :-)

@GuruVishnu What if a sphere is placed on frictionless floor and we give an impulse to it, would it start performing pure rolling after some time?

@DeepamSarmah for the sphere to start rolling its angular acceleration has to be non-zero and that means there has to be a non-zero toque acting on t. Yes?

Umm.. let me rephrase the question

@DeepamSarmah If the impulsive force doesn't cause the sphere to rotate about its centre of mass, then the sphere only translates without any rotation. We need to define whether any friction exists between the impulsive force's source and the sphere. If there exists any other tangential force, and if causes a non-zero angular speed, then the object would roll. But it may or may not be pure rolling (might slip).

If I give an impulse to the sphere along the horizontal line containing the center, can it start pure rolling?

If the surface is frictionless, then the sphere will just translate with uniform velocity.

Thanks! :)

@DeepamSarmah this is what I'm addressing. To make the sphere rotate you ned to apply a torque, but if the floor is frictionless there is no force between the sphere and floor so the torque is zeo.

Therefore the sphere cannot start rotating.

@JohnRennie Thanks a lot :)

Not an advertisement. But if you're confused on what will happen to a ball kept on a frictionless incline, I'd recommend you to read this answer by Dale:

Quite informative and removes some misconceptions.

@GuruVishnu Thanks, that was really helpful :)

How could Torque be calculated in this problem ?

The answer is given B.

@DeepamSarmah The whole system is rotating about the horizontal axis, so the two weights are moving in a circle at angular velocity $\omega$. Yes?

Yeah

Oh, I got it

:-)

The torque due to centripetal force is the answer right ?

Yes :-)

Nice thanks a lot ! :)

I need to go now, but you're always welcome to ping me and I'll reply when I'm back.

@JohnRennie Thanks a lot sir :)