A spherical oil tank with a radius of 10 feet is half full of oil that weighs 60 pounds per cubic foot. Its top is 6 feet underground. How much work is needed to pump the oil out of the tank through a hole in its top?
How do I solve this? Does the fact that it its top is 6 feet underground change the work needed than if it wasn't underground?
Good grief - English units!
You need to find the volume of the tank when half full
V = (1/2) (4/3) pi r^3
then the weight of oil in the tank
W = 60 (V)
Now how far below the ground surface is the center of gravity of the oil?
You can look here http://www.youtube.com/watch?v=KBGsorVYqfo and find that the center of mass of the oil is (3R/8) below the center of the sphere
So
Distance from ground to CG of oil = 6 +10 + (3*10/8) feet
Work done = weight of oil * distance from ground to CG of oil (in foot pounds)
I assume that you have to pump the oil up to the ground by the way.
To solve this problem, we can break it down into several steps:
Step 1: Find the volume of the oil in the tank.
The volume of a spherical tank can be calculated using the formula V = (4/3) * π * r^3, where r is the radius of the tank.
Plugging in the given radius of 10 feet, we can calculate the volume of the tank:
V = (4/3) * π * 10^3
V = (4/3) * π * 1000
V = 4000π cubic feet
Since the tank is half full of oil, the volume of the oil in the tank can be calculated as:
Volume of oil = 1/2 * 4000π
Volume of oil = 2000π cubic feet
Step 2: Find the weight of the oil in the tank.
The weight of the oil can be calculated by multiplying the volume of the oil by its density:
Weight of oil = Volume of oil * Density of oil
Weight of oil = 2000π * 60
Weight of oil ≈ 120000π pounds
Step 3: Find the work done to pump the oil out of the tank.
To lift the oil out of the tank, we need to overcome the gravitational force acting on it. The work done in pumping the oil out of the tank is equal to the gravitational potential energy of the oil at the top of the tank.
The work done can be calculated using the formula W = m * g * h, where m is the mass, g is the acceleration due to gravity, and h is the height.
In this case, the mass of the oil is equal to its weight, and the height is the distance from the top of the tank to the ground level.
Given that the top of the tank is 6 feet underground, the height is 6 feet.
So, the work done is:
W = Weight of oil * g * h
W = 120000π * 32.2 * 6
W ≈ 733008π foot-pounds
Therefore, the work needed to pump the oil out of the tank through the hole in its top is approximately 733008π foot-pounds.
Regarding whether the fact that the top of the tank is underground affects the work needed, it does make a difference. Since the tank top is underground, the oil needs to be pumped against the force of gravity not just to the ground level, but also an additional 6 feet further up to reach the hole in the top of the tank. If the tank top were not underground, the work needed would be slightly lower.