Question

A solid disc and a ring, both of radius 10 cm are placed on a horizontal table simultaneously, with initial angular speed equal to 10 π rad s^-1. Which of the two will start to roll earlier? The co-efficient of kinetic friction is μ_k = 0.2.

Answer 1

Radii of the ring and the disc, r = 10 cm = 0.1 m
Initial angular speed, ω₀ =10 π rad s^–1
Coefficient of kinetic friction, μ_k = 0.2
Initial velocity of both the objects, u = 0
Motion of the two objects is caused by frictional force. As per Newton’s second law
of motion, we have frictional force, f = ma μ_kmg= ma Where, a = Acceleration
produced in the objects m = Mass a = μ_kg … (i)
As per the first equation of motion, the final velocity of the objects can be obtained as:

v = u + at
= 0 + μ_kgt
= μ_kgt … (ii)

The torque applied by the frictional force will act in perpendicularly outward direction and cause reduction in the initial angular speed.

Using the first equation of rotational motion to obtain the final angular speed:

Since t_d > t_r, the disc will start rolling before the ring.