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 μk = 0.2.
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, μk
Initial velocity of both the objects, u
The frictional force causes motion between two objects.
As per Newton’s second law of motion,
Frictional force, f
= Acceleration produced in the objects m
As per the first equation of motion,
Final velocity of the objects can be obtained as,v
= 0 + μkgt
The torque applied by the frictional force will act in perpendicularly outward direction and cause reduction in the initial angular speed.
Torque, τ= – I α
α = Angular acceleration
= –I α
∴ α =
Using the first equation of rotational motion to obtain the final angular speed,
ω = ω0
Rolling starts when linear velocity, v
Equating equations (ii)
, we get
For the ring, I
=0.80 s ...(vii)
For the disc:
I = mr2
, the disc will start rolling before the ring.