Two discs of moments of inertia I₁ and I₂ about their respective axes (normal to the disc and passing through the centre), and rotating with angular speed ω₁ and ω₂ are brought into contact face to face with their axes of rotation coincident.

A. Does the law of conservation of angular momentum apply to the situation? why?

B. Find the angular speed of the two-disc system.

C. Calculate the loss in kinetic energy of the system in the process.

D. Account for this loss.

(n-1) equal point masses each of mass m are placed at the vertices of a regular n-polygon. The vacant vertex has a position vector a with respect to the centre of the polygon. Find the position vector of centre of mass.

Find the centre of mass of a uniform A. half-disc, B. quarter-disc.

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 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.

Two cylindrical hollow drums of radii R and 2R, and of a common height h, are rotating with angular velocities ω (anti-clockwise) and ω (clockwise), respectively. Their axes, fixed are parallel and in a horizontal plane separated by (3 R + δ). They are now brought in contact (δ → 0).

A. Show the frictional forces just after contact.

B. Identify forces and torques external to the system just after contact.

C. What would be the ratio of final angular velocities when friction ceases?