Question

Use conservation of energy to determine the angular speed of the spool shown in the figure below after the 3.00-kg bucket has fallen 3.50 m, starting from rest. The light string attached to the bucket is wrapped around the spool and does not slip as it unwinds.

Answer 1

Conservation of energy leads to

when bucket fallen through a distance , gravitational potential energy is converted into
kinetic energy, here kinietic energy is sum of rotational and translationsl kinetic energy
   mgh = (1/2 mv^2 + 1/2 I W^2)
then acceleration of the

a = g[1 - m/(m + 2M)] where M is the falling mass and m is the spool mass.

a = 9.8*[1 - 5/(5+2*3)] = 5.345 m/s²

Vb = √(2ay) = 6.1168 m/s

w = Vb/R = 6.1168/.6 = 10.195 rad/se

angular speed of the spool is 10.915 rad/s