Question

Two discs of moments of inertia `I_(1) and I_(2)` about their respective axes (normal to the disc and passing through the centre) and rotating with angular speeds `omega_(1) and omega_(2)` 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.

Answer 1

(a) By the conservation of angular momentum
`I_(1)omega_(1)+I_(2)omega_(2)=(I_(1)+I_(2))omega`
`omega=(I_(1)omega_(1)+I_(2)omega_(2))/(I_(1)+I_(2))`, where `omega` is the final angular speed
(b) Initial `K.E., K_(i)=(1)/(2)I_(1)omega_(1)^(2)+(1)/(2)I_(2)omega_(2)^(2)`
Final `K.E., K_(f)=(1)/(2)(I_(1)+I_(2))omega^(2)`
`=(1)/(2)(I_(1)+I_(2))((I_(1)omega_(1)+I_(2)omega_(2))/(I_(1)+I_(2)))^(2)`
`K_(i)=K_(f)=(1)/(2)(I_(1)I_(2))/(I_(1)+I_(2))(omega_(1)-omega_(2))^(2)=+ve, K_(i)gtK_(f)`
`K_(i)-K_(f)=DeltaK:` loss in kinetic energy