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Question:

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 = √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 ω. The angular momentum of the entire assembly about the point 'O' is →L (see the figure). Which of the following statement(s) is(are) true?

The magnitude of angular momentum of centre of mass of the assembly about the point O is 81ma²ω

The magnitude of the z-component of →L is 55ma²ω

The centre of mass of the assembly rotates about the z-axis with an angular speed of ω/5

The magnitude of angular momentum of the assembly about its centre of mass is 17ma²ω/2

Solution:

Let the angle of axis with horizontal be θ. cosθ = l/√l² + a² = √24/5
Velocity of point P (center of lower disk) aω = lΩ
Then Ω = aω/l
Angular velocity of C.M. w.r.t z axis = Ωcosθ = ω/5
LD-CM = ma²/2 ω + 4m(2a)²/2 ω = 17ma²ω/2