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Surface Energy

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Question

A small bucket of mass M is attached to a long inextensible cord of length L. the bucket is released from rest when the cord is in a horizontal position. In its lowest position the bucket scoops up m of water, what is the height of the swing above the lowest position?

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Solution

$A t t h e l o w e s t p o s i t i o n t h e v e l o c i t y o f t h e b u c k e t i s \sqrt{2 g L} K E o f t h e b u c k e t i s M g L . T h e n i t s c o o p s s o m e w a t e r s o, i t s w e i g h t b e c o m e s M + m . V e l o c i t y o f t h e b u c k e t j u s t a f t e r i t s c o o p s s o m e w a t e r i s M \sqrt{2 g L} = (M + m) v v = \frac{M \sqrt{2 g L}}{M + m} m a x i m u m h e i g h t r e a c h e d i s \frac{v^{2}}{2 g} = {(\frac{M}{M + m})}^{2} L$

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