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1)

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 centers, This assemply 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 assemply about the point 'O' is L (see the figure). Which of the following statement(s) is (are) true?

1122021137_b63.png


A) The magnitude of the z- component of L is 55ma2ω

B) The magnitude of angular momentum of centre of mass of the assembly about O is 81ma2 ω

C) The centre of mass of the assembly rotates about the Z-axis with an angular speed of ω5

D) The magnitude of angular momentum of the assembly about its centre of mass is 17ma2ω2

Answer:

Option C,D

Explanation:

1122021479_b64.png

cosθ=1l2+a2=245

(c) Velcocity of point P:aω=1Ω then

Ω = aω1= Angular velocity of C M w.r.t

Point O. Angular velocity fo CM w.r.t Z - axis = Ω cosθ

ωCMZ=aω1245=aω24a245

ωCMZ=aω5

(a) Lz=LCM0cosθLDCMsinθ

=81245a2mω×24517ma2ω2×124

=81×24ma2ω2517ma2ω224

(b) LCMO=(5m)[9l5Ω]9l5=81ml2ω5

  =81ml25×aωl

LCMO=81mlaω5=8124a2mω5

(d)  LDCM=ma22ω+4m(2a)22ω=17ma2ω2