A thin uniform circular disc of mass M and radius R is rotating in a horizonatl plane about an axis passing through its centre and perpendicular to its plane with an angular velocity `omega`. Another disc of same thickness and radius but of mass `(1)/(8)` M is placed gently on the first disc coaxially. The anuglar velocity of the system is now

Options

A. `(8)/(9) omega`
B. `(5)/(9) omega`
C. `(1)/(3) omega`
D. `(2)/(9) omega`

Answer

a

Explanation

Given, mass of first circular disc `(m_(1)) = M`
Radius of first circular disc `(r_(1)) = R`
Mass of second circular disc `(m_(2)) = (1)/(8) M`
For first disc,
`I_(1) = (MR^(2))/(2)`
`= implies L_(1) = (MR^(2) omega_(1))/(2)`
For first and second disc system,
`I_(2) = (MR^(2))/(2) + (M)/(8) . (R^(2))/(2)`
or `I_(2) = (9 MR^(2))/(8 xx 2)`
`implies L_(2) = (9 MR^(2))/(8 xx 2) omega_(2)`
Now from conservation of angular momentum,
`L_(1) = L_(2)`
`implies (MR^(2) omega_(1))/(2) = (9 MR^(2))/(8 xx 2) omega_(2) implies omega_(1) = (9)/(8) omega_(2)`
Given `omega_(2) = omega`
`:. omega_(2) = (8)/(9) omega_(1) = (8)/(9) omega`