Question

A bucket of water with a mass of 4.17 kg is attached to a rope that is wound around cylinder. the cylinder has a mass of 7.19 kg is mounted horizontally on frictional bearings. the bucket is released what is its speed after falling distance of .645 m
Don't you mean frictionLESS bearings?
If they are frictional, you have to know how much friction there is to predict the speed.

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