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Question

A disc of mass M radius R is fixed about an axis as shown in figure. If a particle of mass M4 moving with speed v perpendicular to the disc plane, hits the disc at point P and sticks to the disc, the disc starts rotating about its axis as shown. Find the angular velocity of the system (disc + particle) after the hit.


A
ω=vR
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B
ω=v5R
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C
ω=v2R
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D
ω=v5R
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Solution

The correct option is C ω=v2R
As we know from angular momentum conservation.
L=mvr=Iω
MOI of the disc about an axis passing through its centre and perpendicular to its plane is I=MR22
Hence by perpendicular axis theorem, MOI of the disc about given axis,
I=MR24
where I is the MOI about an axis passing through disc centre and in the plane of disc.
Initial angular momentum =mvr
Li=M4.v.R
Final angular momentum of the system
Lf=Iω
Lf=(MR24+(M4)R2)ω
Lf=MR22ω
Hence, from angular momentum conservation
Li=Lf
MvR4=(MR22)ω
i.e ω=v2R

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