A spherical tank of radius 100ft is full of gasoline weighing 40 pounds per cubic feet how much work is done in pumping the gasoline to the top of the tank?
why bother? The tank is already full.
But, let's say that you want to pump all the gas up to the top of the tank
total weight: 4/3 π r^3 * density = 167,551,608.91 lbs
moving the center of mass up 100 ft will take 16,755,160,891 ft-lbs of work
You can use calculus and integrate, moving the weight of all the thin slices of gasoline their respective distances, but it will come out the same.
To calculate the work done in pumping the gasoline to the top of the tank, we need to find the weight of the gasoline and the vertical distance it needs to be lifted.
The weight of an object can be calculated by multiplying its volume by its density. In this case, the volume of the gasoline will be the volume of a spherical segment, which is given by the formula:
V = (1/6)πh(3a^2 + h^2)
where V is the volume of the spherical segment, h is the height of the segment, and a is the radius of the sphere.
In this case, the radius of the tank is 100ft, so a = 100ft.
The height of the segment is equal to the height of the tank, which is the same as the radius (100ft).
So, plugging in the values in the formula, we have:
V = (1/6)π(100ft)(3(100ft)^2 + (100ft)^2)
≈ 196,349 ft^3
Now, we can calculate the weight of the gasoline by multiplying the volume by the density:
Weight = V * density
= 196,349 ft^3 * 40 lb/ft^3
= 7,853,960 lb
Since the gasoline needs to be pumped to the top of the tank, the vertical distance it needs to be lifted is equal to the height of the tank, which is 100ft.
The work done in pumping the gasoline is given by the formula:
Work = Force * Distance
In this case, the force is equal to the weight of the gasoline, which is 7,853,960 lb, and the distance is the height of the tank, which is 100ft.
So, plugging in the values in the formula, we have:
Work = 7,853,960 lb * 100ft
= 785,396,000 ft-lb
Therefore, the work done in pumping the gasoline to the top of the tank is approximately 785,396,000 ft-lb.