The reel shown in the figure above has radius R and moment of inertia I. One end

Question

The reel shown in the figure above 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 unstretched position and the reel is then released from rest. Find the angular speed of the reel when the spring is again unstretched. (Use any variable stated above along with the following as necessary: g for the acceleration of gravity, and ?.) ? = Worked on this one for an hour, I get the same equation approaching it both from kinetic equations and energy equations... I'm lost. The reel shown in the figure above 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 unstretched position and the reel is then released from rest. Find the angular speed of the reel when the spring is again unstretched. (Use any variable stated above along with the following as necessary: g for the acceleration of gravity, and ?.) ? = Worked on this one for an hour, I get the same equation approaching it both from kinetic equations and energy equations... I'm lost.

Explanation / Answer

find final angular velocity w initial energy = kd^2/2 final kinetic energy = Iw^2/2 + mv^2/2 = Iw^2/2 + m(wR)^2/2 = (I + mR^2)w^2/2 final potential energy = -mgdsin(theta) energy conservation: kd^2/2 = (I + mR^2)w^2/2 - mgdsin(theta) so w = sqrt[(kd^2 + 2mgdsin(theta))/(I + mR^2)]

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