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?

Answer

C, D

Explanation

(i) `L_(z) = I_(CM-0) cos theta -L_(D-CM) sin theta`
`=(81sqrt(24))/25 a^(2) m omega xx sqrt(24)/5 -(17 ma^(2) omega)/2 xx 1/sqrt(24)`
`=(81 xx 24 m a^(2) omega)/25 - (17 ma^(2) omega)/(2sqrt(24))`
(ii) `L_(CM-0) = (5m) [(91)/5 omega](9l)/5 = (81 mI^(2) omega)/5, =(81 m I^(2))/5 xx (a omega)/I`
`L_(CM-0) =(81 mlaomega)/5 =(81 sqrt(24) a^(2) momega)/5`
(iii) Velocity of point `P:a omega = 1 omega` then,
`omega =(aomega)/1` =Angular velocity of C.M. w.r.t. point O.
Angular velocity of CM w.r.t. Z-axis = `omega cos theta`
`omega_(CM-z) = (aomega)/1 sqrt(24/5) = (aomega)/sqrt(24a) sqrt(24)/5, omega_(CM-z) = (aomega)/5`
(iv) `L_(D-CM) = (ma^(2))/2 omega + (4m(2a)^(2))/2 omega =(17 m a^(2) omega)/2`