Two thin circular discs of mass m and 4m, having radii of a and 2a, respectively, are rigidly fixed by a massless, rigid rod of length `l= sqrt(24a)` through their centres. This assembly is laid on a firm and flat surface, and set rolling without slipping on the surface so that the angular speed about the axis of the rod is `omega.` The angular momemtum of the entire assembly about the point 'O' is `vacL` (see the figure). Which of the follwing statement (s) is (are) true?

Options

A. The magnitude of angular momentum of the assembly about its centre of mass is `17 ma^(2) omega//2`
B. The centre of mass of the assembly rotates about the Z-axis with an angular speed of `omega//5`
C. The magnitude of the Z-component of `vecL` is `55 ma^(2)omega`
D. The magnitude of angular momentum of centre of mass of the assembly about the point is `81 ma^(2) omega`

Answer

A, B

Explanation

It is a case of pure rotation about the centre of mass hence angular momentum of the system about its centre of mass can be written as
`L= I omega`
`L=[(ma^(2))/(2)+(4m (2a)^(2))/(2)] omega]`
`L=(117 ma^(2)) omega)/(2)`
Hence option (a) is correct.
`OB=X sqrt(l^(2)+a^(2))`
`X=a sqrt(24+1)`
`X=5a`
Velocity of Disc A in terms of 2, can be written as
`V_(A)= Omega_(z)X=5a Omega_(z)`
Disc performs pure rolling `rArr V_(A)= omega a`
`5a Omega_(z)= omega a`
Omega_(z)=(omega)/(5)`
Hence option (b) is correct.