A cylinder at rest lying on a rough ground is given an impulse which imparts a translational velocity (no angular velocity) to it. The question goes ahead with finding time after which rolling starts etc. etc.
But I have another question: The friction from the ground will decrease the translational velocity but increase the angular velocity (as the torque of friction will be in the direction of rolling). After some time, the translational velocity will become equal to angular velocity and hence pure rolling will start.
After this time work done by friction will be 0 as the point of contact is not moving and hence translational velocity will be constant. Now my doubt is that, when translational velocity is constant, it implies angular velocity will also be constant (as translational velocity is a multiple of angular velocity). But the torque from friction (about the centre of mass) will not be zero which means angular velocity will not be constant. In some book where there was the same problem it was written friction will vanish. How can friction vanish? How can anguluar velocity be constant even when there is a torque by friction?