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The reel shown in figure has radius R and moment of inertia I. One end of the block of mass m is connected to a spring of force constant k and the other end is fastened to a cord wrapped around the reel. The reel's axle and incline are frictionless. The reel is wound counterclockwise so that the spring stretches a distance d from its unstretched position and the reel is then released from rest. The angular speed of the reel when the spring is again unstretched is

A
2mgdsinθ+kd2I + mR2
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B
mgdsinθ+kd2I + mR2
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C
2mgdsinθkd2I + mR2
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D
mgdsinθkd2I + mR2
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Solution

The correct option is A 2mgdsinθ+kd2I + mR2
The block and end of the spring are pulled a distance d up the incline and then released. The angular speed of the reel and speed of the block are related by v=ωR. The block-reel-earth system is isolated, so
ΔK+ΔU=0.
(mv220)+(Iω220)+(0mgdsinθ)+(0kd22)=0
ω22(I+mR2)=mgdsinθ +kd22
ω=2mgdsinθ+kd2I + mR2

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