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 stretches a distance d from its $$un-stretched$$ position and the reel is then released from rest. The angular speed of the reel when the spring is again $$un-stretched$$ is

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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 $$=$$ ωR. The $$block-reel-Earth$$ system is isolated, so∆K $$+$$∆U $$=$$ $$0(12mv2-0)+(12I$$ω$$2-0)+(0-mgd$$ $$sin$$ θ$$)+(0-12kd2)$$ $$=$$ $$012$$ω$$2(I+mR2)$$ $$=$$ mgd $$sin$$ θ $$+$$ $$12kd2$$⇒ω $$=$$ $$2mgd$$ $$sin$$ θ $$+kd2I$$ $$+$$ $$mR2$$

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