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

(i) Let w be the angular speed of the two-disc system. Then by conservation of angular momentum.
`(I_(1) + I_(2))omega = I_(1)omega_(1) + I_(2)omega_(2)`
Or `omega =(I_(1)omega_(1) + I_(2)omega_(2))/(I_(1) + I_(2))`
(ii) Initial K.E. of the two discs.
`K_(1) = 1/2 I_(1)omega_(1)^(2) + 1/2 I_(1)omega_(2)^(2)`
Final K.E. of the two disc system.
`K_(2) = 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)`
Loss in K.E.
`= K_(1)-K_(2) =1/2(I_(1)omega_(1)^(2) + I_(2)omega_(2)^(2)) -1/(2(I_(1) + I_(2)) (I_(1)omega_(1)^(2) + I_(2))(omega_(1) - omega_(2))^(2)` = a positive quantity `[therefore omega_(1) ne omega_(2)]`
Hence there is a loss of rotational K.E. which appears as heat.
When the two discs are brought together, work is done against friction between the two discs.