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 axle and the incline are frictionless. The reel is wound counterclockwise so that the spring strectches 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.
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Options

A. `sqrt((2 mgd sin theta + kd^2)/(I + mR^2))`
B. `sqrt((mgd sin theta + kd^2)/(I + mR^2))`
C. `sqrt((mgd sin theta - kd^2)/(I + mR^2))`
D. `sqrt((2 mgd sin theta - kd^2)/(I + mR^2))`

Answer

A

Explanation

(a) 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 the speed of the block are related by `v omega R`. The block-reel-Earth system is isolated, so
`Delta K + Delta U = 0`
`((1)/(2) mv^2 - 0) + ((1)/(2) I omega^2 -0)`
`+(0 = mgd sin theta) + (0 - (1)/(2) kd^2) = 0`
`(1)/(2) omega^2(I + mR^2) = mgd sin theta + (1)/(2) kd^2`
`rArr omega = sqrt((2 mgd sin theta + kd^2)/(I + mR^2))`.
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