A bucket that weighs 3.2 pounds and a rope of negligible weight are used to draw water from a well that is 82 feet deep. The bucket is filled with 42 pounds of water and is pulled up at a rate of 2.3 feet per second, but water leaks out of a hole in the bucket at a rate of 0.35 pounds per second. Find the work done pulling the bucket to the top of the well.
The net mass of the bucket as a function of time, M(t)
=3.2+42-0.35t
Work done as a function of time
= M(t)*2.3 ft/s * 32.2 ft/s^2
Over the height of the well,
time required to go to the top
= 82'/2.3 '/s
= 82/2.3 s
Total work done
= ∫ M(t)*32.2 * 2.3dt from 0 to 82/2.3
=[45.2t-0.175t²] from 0 to 82/2.3
= 102872 poundal-ft
= 3194.8 lbf-ft (after dividing by g)
what's the 32.2ft/s^2
Acceleration due to gravity
To find the work done pulling the bucket to the top of the well, we can calculate the total work done against gravity and subtract the work done by the leaking water.
First, let's calculate the work done against gravity. The work done against gravity is given by the formula:
Work = Force * Distance
In this case, the force is the weight of the bucket and water, and the distance is the height of the well. The weight of the bucket and water is the sum of the weight of the bucket and the weight of the water, which is calculated as follows:
Weight of the bucket = 3.2 pounds
Weight of the water = 42 pounds
Total weight = Weight of the bucket + Weight of the water = 3.2 + 42 = 45.2 pounds
Now, let's convert the height of the well from feet to pounds. Since we know that 1 pound is equivalent to 1 foot, the height of the well in pounds is 82 pounds.
Now we can calculate the work done against gravity using the formula:
Work against gravity = Total weight * Height of the well
Work against gravity = 45.2 * 82
Next, let's calculate the work done by the leaking water. The work done by the leaking water is the force of the water leaking multiplied by the distance it travels. The force of the leaking water is the rate at which the water leaks, which is given as 0.35 pounds per second. The distance it travels is the same as the distance the bucket is pulled up, which is given as 82 feet.
Work done by leaking water = Force of leaking water * Distance traveled
Work done by leaking water = 0.35 * 82
Finally, let's subtract the work done by the leaking water from the work done against gravity to find the total work done pulling the bucket to the top of the well:
Total work done = Work against gravity - Work done by leaking water
Total work done = (45.2 * 82) - (0.35 * 82)
After calculating this equation, you will find the solution to the problem, which is the work done pulling the bucket to the top of the well.