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Question

Two coaxial discs, having moments of inertia I1 and I1/2 are rotating with respective angular velocities ω1 and ω1/2, about their common axis. They are brought in contact with each other, and thereafter they rotate with a common angular velocity. If Ef and Ei, are the final and initial total energies, then (EfEi) is

A
I1ω2124
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B
I1ω2112
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C
38I1ω21
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D
I1ω216
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Solution

The correct option is A I1ω2124
Initial kinetic energy of the given system,

Ei=12I1ω21+12(I12)(ω12)2

Ei=916I1ω21 .........(i)

Net external torque on the system is zero, so using law of conservation of angular momentum.

Initial angular momentum = Final angular momentum

I1ω1+(I12)(ω12)=I1ω+I12ω

(assuming angular velocity after contact is ω)

54ω1=32ω

ω=56ω1 ..........(ii)

Now, final kinetic energy (after contact) is

Ef=12I1ω2+12(I12)ω2=34I1ω2

Putting the value of ω from eq. (ii)

Ef=34I1(56ω1)2=2548I1ω21

Hence, change in KE,
ΔE=EfEi=2548I1ω2I916I1ω21

EfEi=124I1ω21

Hence, option (A) is correct.

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