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Question

If two discs of moments of inertia I1 and I2, rotating about a collinear axis passing through their centres of mass and perpendicular to their planes with angular speeds ω1 and ω2 respectively in opposite directions are made to rotate combinedly along the same axis, then the magnitude of new angular velocity of the system is

A
I1ω1+I2ω2I1+I2
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B
I1ω1I2ω2I1+I2
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C
I1ω2+I2ω1I1+I2
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D
I1ω1I2ω2I1I2
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Solution

The correct option is B I1ω1I2ω2I1+I2
Consider both discs as a system.
Since there is no external net toreque on the system, angular momentum about the rotational axis will be conserved.
Initial angular momentum (Li)=I1ω1I2ω2
(-ve because both are rotating in opposite directions)
Assume ω will be the common angular velocity when both are rotated combinedly.
Then, final angular momentum Lf=I1ω+I2ω
Lf=(I1+I2)ω
(because both are rotating in same direction)
From Li=Lf
I1ω1I2ω2=(I1+I2)ω
i.e ω=I1ω1I2ω2I1+I2

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