A 3-lb bucket containing 20- lb of water is hanging at the endof a 20-ft rope that weights 4 -oz / ft. The other end of the ropeis aattached to a puley. How mauch work is required to wind thelenght of the rope onto the pulley, assuming that the rope iswound onto the pulley at a rate of 2- ft/s and that as the bucketis being lifted, water leaks from the bucket at a rate of 0.5 -lb/s?
340
To determine the amount of work required to wind the length of the rope onto the pulley, we need to consider the forces acting on the system and calculate the work done against these forces.
Let's break down the problem step by step:
1. Calculate the weight of the water:
The water weighs 20 lb, so the weight of the water is 20 lb.
2. Calculate the weight of the rope:
The rope weighs 4 oz per foot, and the rope is 20 ft long. To convert the weight to pounds, we need to divide by 16 (since there are 16 ounces in a pound):
Weight of the rope = (4 oz/ft) * (20 ft) / 16 = 5 lb
3. Calculate the net force acting on the system:
The net force is the difference between the weight of the water and the weight of the rope since these forces act in opposite directions:
Net force = weight of water - weight of rope = 20 lb - 5 lb = 15 lb
4. Calculate the work done against the net force:
Work is calculated by multiplying force by distance. In this case, the distance is the length of the rope being wound onto the pulley, and the force is the net force:
Work = force * distance = 15 lb * 20 ft = 300 ft-lb
5. Account for the water leak:
Given that water leaks from the bucket at a rate of 0.5 lb/s, we need to consider the time it takes to wind the rope onto the pulley. We can calculate the time based on the rate of rope winding:
Time = Distance / Speed = 20 ft / 2 ft/s = 10 s
Now, we need to account for the water leak over this time period:
Water leaked = leak rate * time = 0.5 lb/s * 10 s = 5 lb
6. Adjust the work done due to water leakage:
Since the leaked water reduces the weight being lifted, the actual work done against the net force will be less. We subtract the weight of the leaked water from the net force:
Adjusted net force = net force - water leaked = 15 lb - 5 lb = 10 lb
7. Recalculate the work done against the adjusted net force:
Work = force * distance = 10 lb * 20 ft = 200 ft-lb
So, the work required to wind the length of the rope onto the pulley, accounting for the water leakage, is 200 foot-pounds (ft-lb).