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 ?

Explanation

(a) Let `l_(1) " and " l_(2)` be the moments of inertia of two discs having angular speeds `w_(1) " and " w_(2)` respectively. When they are brought in contact, the moment of inertia of the two-disc system will be `l_(1)+l_(2)`. Let the system now have an angular speed w. From the law of conservation of angular momentum, we know that
`I_(1)omega_(1)+I_(2)omega_(2)=(I_(1)+I_(2))omega`
`therefore` The angular speed of the two-disc system,
`omega=(I_(1)omega_(1)+I_(2)omega_(2))/(I_(1)+I_(2))`
(b) The sum of kinetic energies of the two discs before coming in contact,
`k_(1)=(1)/(2)I_(1)omega_(1)^(2)+(1)/(2)I_(2)omega_(2)^(2)`
The final kinetic energy of the two-disc system,
`k_(2)=(1)/(2)(I_(1)+I_(2))omega^(2)`
`=(1)/(2)(I_(1)+I_(2))xx((I_(1)omega_(1)+I_(2)omega_(2))/(I_(1)+I_(2)))^(2)`
`=(1)/(2) ((I_(1)omega_(1)+I_(2)omega_(2))/(I_(1)+I_(2)))^(2)`
Now, `k_(1)-k_(2)=(1)/(2)I_(1)omega_(1)^(2)+(1)/(2)I_(2)omega_(2)^(2)-(1)/(2)((I_(1)omega_(1)+I_(2)omega_(2))/(I_(1)+I_(2)))^(2)`
`=(1)/(2(I_(1)+I_(2))xx[(I_(1)omega_(1)^(2)+I_(2)omega_(2)^(2))(I_(1)+I_(2))-(I_(1)^(2)omega_(1)^(2)+I_(2)^(2)+2I_(1) I_(2)omega_(1)omega_(2))]`
`(1)/(2(I_(1)+I_(2)))xx[I_(1)^(2)omega_(1)^(2)+I_(2)^(2)+I_(2)^(2)+omega_(2)^(2)+I_(1)I_(2)omega_(1)^(2)+I_(1)I_(2)omega_(2)^(2)-I_(1)^(2)omega_(1)^(2)-I_(2)^(2)omega_(2)^(2)-2I_(1)I_(2)omega_(1)omega_(2)]`
`(1)/(2(I_(1)+I_(2))[I_(1)I_(2)(omega_(1)^(2)+omega_(2)^(2)-2omega_(1)omega_(2))]`
`=(I_(1)I_(2))/(2(I_(1)+I_(2)))(omega_(1)-omega_(2))^(2)`
Now, `(w1-w2)`2 will be positive whether w1 is greater or smaller than w2.
Also, `l_(1)l_(2)//2(l_(1)+l_(2))` is also positive because `l_(1) " and l_(2)` are positive.
Thus, `k_(1)-k_(2)` is a positive quantity.
`because k_(1)=k_(2)+a` positive quantity or `k_(1) gt k_(2)`
`because` The kinetic energy of the combined system `(k_(2))` is less than the sum of the kinetic energies of the two dies. The loss of energy on combining the two discs is due to the energy being used up because of the frictional forces between the surfaces of the two discs. These forces, in fact , bring about a common angular speed of the two discs on combining.