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A disc of radius R is rotating with an angular speed ω0 about a horizontal axis. It is placed on a horizontal table. The coefficient of kinetic friction is μk.

(a) What was the velocity of is its centre of mass before being brought in contact with the table?

(b) What happens to the linear velocity of a point on its rim when placed in contact with the table?

(c) What happens to the linear speed of the centre of mass when disc is placed in contact with the table?

(d) Which force is responsible for the effects in (b) and (c)?

(e) What condition should be satisfied for rolling to begin?

(f) Calculate the time taken for the rolling to begin.

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(a) Before being brought in contact with the table the disc was in pure rotational motion (only) about its axis passing through centre –

∴ vcm = 0 as the point on axis are considered at rest.

(b) When the rotating disc is placed in contact with table’s surface due to friction, linear velocity of a point on the rim decreases.

(c) When disc is placed in contact with the surface of the table, due to frictional force, centre of mass acquires some linear velocity. The linear velocity increases.

(d) Frictional force

(e) When rolling of disc starts on table, then vcm = ωR due to reaction force.

(f) Acceleration produced in centre of mass due to reaction force F because of frictional force in disc

Angular retardation produced by the torque due to friction.

Here, frictional force helps in rolling motion.

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