In an old fashion well, a basket is suspended over well shaft by a winch and rop

Question

In an old fashion well, a basket is suspended over well shaft by a winch and rope. The winch includes a solid cylinder with mass M and radius R and rotates without friction about a horizontal axis. The bucket ( mass m) must descend a height h to reach the water; it is suspended by a rope of negligible mass that wraps around the winch. if the winch handle fall off, releasing the bucket to fall to the water, rotating the cylinder as it falls, fine the speed v of the bucket and the angular velocity w of the cylinder just before the bucket hits the water. please show work

Explanation / Answer

Initial P.E = m*g*h
Initial K.E = 0

Final P.E = 0
Final K.E = 1/2 * mv^2
Final Rotational Energy = 1/2 I*w^2

Where w = v/R
I = 1/2 * M*R^2

Final Rotational Energy = 1/2 * 1/2 * M*R^2 (v^2/R^2)
Final Rotational Energy = 1/4 * Mv^2

Using Enery Conservation -

Initial P.E + Initial K.E = Final K.E + Final P.E + Final Rotational Energy
m*g*H + 0 = 1/2 * mv^2 + 0 + 1/4 * Mv^2
v^2 (1/2 m + 1/4 M) = m*g*h
v^2 = m*g*h/ (1/2 m + 1/4 M)
v = sqrt(m*g*h/ (1/2 m + 1/4 M) )

Angular Velocity, w = v* R
w = sqrt(m*g*h/ (1/2 m + 1/4 M) ) * R

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