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 speed `omega_(1) and omega_(2)` are brought into contact face to face with their axes of rotation coincident . What is the angular speed of the two-disc system ?

Options

A. Does the law of conservation of angular monentum 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 the process.
D. Account for this loss.

Explanation

Consider the diagram below
let the common angular velocity of the system is `omega `
( A) yes , the law of conservation of angular momentum can be applied ,Because , there is no net external torque on the system of the two disc .
External forces , gravition and normal reaction ,act through th axis of rotation , hence produce no torque .

( B) BY conservation of angular momentum
`L_(f)=L_(j)`
`implies Iomega=Iomega_(1)+I_(2)omega_(2)`
` implies omega=(I_(1)+I_(2)omega_(2))/(I)=(I_(1)omega_(1)+I_(2)omega_(2))/(I_(1)+I_(2))" "(:' I=I_(1)+I_(2))`
`(C) K_(f)=(1)/(2)(I_(1)+I_(2))((I_(1)omega_(1)+I_(2)omega_(2)))/((I_(1)+I_(2)))=(1)/(2)((I_(1)omega_(1)+I_(2)omega_(2)))/((I_(1)+I_(2)))`
`k_(f)=(1)/(2)(I_(1)omega_(1)^(2)+I_(2)omega_(2)^(2))`
`DeltaK=K_(f)-K_(i)=-(I_(1)I_(2))/(2(I_(1)+I_(2)))(omega_(1)-omega_(2))^(2)lt0`
(D) Hence there is loss in KE of the loss in kintic energy is mainly due to the work against the frication between the two discs.