Question

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 (3R + 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?

Answer 1

(a)

Let us consider the following figure, which shows frictional forces.

The direction of v₁ and v₂ at point of contact are tangentially upward.

Then

⇒ v₁ = ωR, v₂ = ω2R

(b) External forces acting on the system are equal and opposite, so net force is zero. F’ = F = F” where F’ and F” are external force through support

F_net = 0

External torque = F × 3R, anticlockwise.

So, as velocity of drum 2 is double, i.e., v₂ = 2v₁

(c) Let ω₁ and ω₂ be final angular velocities (anticlockwise and clockwise respectively) of smaller drum 1 and drum 2 respectively. finally, when their velocities become equal there will be no friction due to no slipping at this stage.

Hence, Rw₁ = 2w₂ or w₁/w₂ = 2/1