A cylindrical tank of 22.1m high with radius 12.2m is filled to 10m high with water. How much work must be done to pump all the water out of the tank?
(density of water is 1000\ kg/m^3)
I used similar triangles somewhere in my calculations to find the answer, but I'm not sure I'm doing it right...
I got some big number like 1483087pi.
What similar triangles? The tank has a circular cross-section of πr^2
Work = force (weight)*distance, so figure the weight of a thin slice of water (thickness dy).
Then the work done to lift it over the top from depth y is its weight times y. Add up all those work pieces by integrating from 11.1 to 22.1, the range of depths (distance from the top of the tank).
LOL, consider it frozen :)
Well, it would be easier to siphon it out but anyway:
How much work must you do to lift that amount of water from the height of its center of mass in the tank to the top of the tank?
Volume of water = pi r^2(10)
= 10 pi (12.2^2) m^3
center of gravity of water above ground 5 m
top of tank = 22.1 m
so raise the water 22.1 - 5 = 17.1 meters
work done = m g h
= 1000kg/m^3 * 10 pi (12.2^2) * 9.81 * 17.1 Joules
Now you could have done that layer by layer integrating, but really no need.
I suppose you posted that question just to find out who here was a mathematician and who an engineer :)
Nice work, Damon. I like how the engineers always cut to the heart of the problem.
Reminds me of the engineer's way to measure the area of an irregular shape. He traced it onto paper, cut it out, and weighed it. Then he weighed 1 cm^2 of paper, and voila! The area was easy.
Just as a check,
∫[12.1,22.1] 1000 π (12.2^2)*9.8 x dx = 7.83595*10^8
which agrees with Damon's number, as expected.
To find the work required to pump all the water out of the tank, we need to calculate the weight of the water first.
1. Find the volume of water in the tank:
Since the tank is cylindrical, we can use the formula for the volume of a cylinder:
Volume = π * r^2 * h
where π is Pi (approximately 3.14), r is the radius, and h is the height.
In this case, the radius (r) is 12.2 meters and the height (h) is 10 meters.
So, the volume of water in the tank is:
Volume = 3.14 * (12.2^2) * 10
2. Calculate the weight of the water:
The weight of an object is given by the formula:
Weight = mass * gravity
where mass is volume * density and gravity is the acceleration due to gravity (approximately 9.8 m/s^2).
In this case, the density of water is 1000 kg/m^3.
So, the weight of the water is:
Weight = (Volume * density) * gravity
3. Calculate the work done to pump the water out:
The work done to pump a liquid out is given by the formula:
Work = force * distance
where force is weight and distance is the height of the water column.
In this case, the distance is the height of the water column in the tank (10 meters) and the force is the weight of the water calculated in step 2.
So, the work done to pump all the water out is:
Work = Weight * distance
Now, let's perform the calculations:
Step 1:
Volume = 3.14 * (12.2^2) * 10
Step 2:
Weight = (Volume * density) * gravity
Step 3:
Work = Weight * distance
By plugging in the values for the radius, height, density, and gravity, you should be able to find the correct answer.