An old oaken bucket of mass 6.75 kg hangs in a well at the end of a rope. The rope passes over a frictionless pulley

frictionless-pulley

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An old oaken bucket of mass 6.75 kg hangs in a well at the end of a rope. The rope passes over a frictionless pulley at the top end of a well, and you pull horizontally on the end of the rope to raise the bucket slowly at the distance of 4.00m.
(a) How much work do you do on the bucket in pulling it up?
(b) How much work does the pulley do on the bucket?
© What is the total work done on the bucket?

Answer:

Now let us first analyse this situation. There are two forces acting on the body, gravitational (mg) downwards and tension from the rope upwards. Now since you’ve said the bucket moves slowly that means it is equilibrium. Thus all forces add up to zero or we can say that tension from the rope equals mg in upward direction. Now let us consider myself. I have rope pulling me with same tension since rope is massless. So by Newton’s third law I pull the rope with equal force. Also there is friction acting on me from the ground in opposite direction as tension but with equal magnitude since I also have to move slowly ( or be in equilibrium) to move the bucket slowly. Now consider the pulley. Does it apply any force? Yes it does. It has to apply some normal force on the rope particle touching it to keep it in equilibrium. This will be 2–√ times the tension if the pulley circular in nature. But you can ignore this detail for this particular question atleast. So if the pulley applies force, does it do some work on that rope particle it applies force on? No it doesn’t. This because the force is normal in nature, and thus it is perpendicular to the motion of particle. Such perpendicular forces don’t do any work.

(a) I actually do my work on the rope particle that touches me or apply force on. This because a force always works on the body it acts upon. This work will be equal to the work tension does on me since these forces are equal and opposite and are associated with equal displacements.Now the catch here is that in a system, all tensions from massless ropes do zero total work. So the work done by tension on bucket is negative of work done by tension on me which itself is negative of work done by me on rope particle. Work done by tension will be the change in potential energy of bucket (or negative of work done by mg on bucket) which is equal to mgh = 264.6 joules taking g = 9.8m/s^2. So work done by me is 264.6 joules.

(b) Pulley as discussed above does no work.
© Total work done on the bucket is zero. Remember total work done is always equal to change in kinetic energy which is zero. Also we can see tension and mg are equal and opposite and cause same displacement, so they do zero net work.