A 3 lb bucket containing 40 lb of water is hanging at the end of a 20 ft rope that weighs 4 oz/ft. The other end of the rope is attached to a pulley. How much work is required to wind the length of rope into the pulley, assuming that the rope is wound onto the pulley at a rate of 2 ft/s and that as the bucket is being lifted, water leaks from the bucket at a rate of 1 lb/s?

Question

A 3 lb bucket containing 40 lb of water is hanging at the end of a 20 ft rope that weighs 4 oz/ft. The other end of the rope is attached to a pulley. How much work is required to wind the length of rope into the pulley, assuming that the rope is wound onto the pulley at a rate of 2 ft/s and that as the bucket is being lifted, water leaks from the bucket at a rate of 1 lb/s?

Answer 1

To determine the work required to wind the length of rope into the pulley, we need to consider the forces at play and calculate the total work done.

First, let's break down the problem into its components and find the relevant values:

1. The weight of the water in the bucket is 40 lb.
2. The weight of the bucket itself is 3 lb.
3. The length of the rope is 20 ft.
4. The weight of the rope is given as 4 oz/ft, which is equivalent to (4/16) lb/ft or 0.25 lb/ft.
5. The rate at which the rope is wound into the pulley is given as 2 ft/s.
6. The rate at which the water leaks from the bucket is given as 1 lb/s.

Now, let's calculate the work required step by step:

1. Calculate the total weight hanging at the end of the rope:
Total weight = Weight of water + Weight of bucket
= 40 lb + 3 lb
= 43 lb

2. Calculate the weight of the rope:
Weight of rope = Weight per foot * Length of rope
= 0.25 lb/ft * 20 ft
= 5 lb

3. Calculate the net force exerted on the rope:
Net force = Total weight - Weight of rope
= 43 lb - 5 lb
= 38 lb

4. Calculate the work done to wind a unit length of rope into the pulley:
Work done per unit length = Net force * Distance
= 38 lb * 1 ft
= 38 ft-lb/ft

5. Calculate the work done per second to wind the length of rope into the pulley:
Work done per second = Work done per unit length * Rate of winding
= 38 ft-lb/ft * 2 ft/s
= 76 ft-lb/s

6. Calculate the work done to lift the leaking water:
Work done to lift the leaking water = Weight of water * Distance
= 40 lb * 1 ft
= 40 ft-lb/s

7. Calculate the total work done:
Total work done = Work done to wind the rope + Work done to lift the water
= 76 ft-lb/s + 40 ft-lb/s
= 116 ft-lb/s

Therefore, the work required to wind the length of rope into the pulley is 116 ft-lb/s.