Posted by Amy on Wednesday, November 10, 2010 at 6:05pm.
An underground tank full of water has the following shape:
Hemisphere - 5 m radius. at the bottom
Cylinder - radius 5 m and height 10m in the middle
Circular cone radius 5 m and height 4 m at the top
The top of the tank is 2 m below the ground surface and is connected to the surface by a spout. find the work required to empty the tank by pumping all of the water out of the tank up to the surface.
density of water = 1000 kg/m^3
Gravity = 10 m/s^2
I am doing to where I have three parts to this question. I find the work of all of them then add the work done of all 3 together.. However, I cannot figure out how to find the work done for the hemisphere OR the circular cone. Please help me solve this out I have no idea where to start!
- Calculus 2 - Damon, Wednesday, November 10, 2010 at 7:00pm
OK, Here is the hemisphere.
we have a hemisphere with base 16 feet below ground and bottom 21 feet below ground.
We need its volume and the distance of the cg below ground.
The volume is easy, half a sphere
(1/2) (4/3) pi r^3 = (2/3) pi 125 = 250 pi/3
the centroid of a sphere is 3/8 r from the base as derived here:
Therefore the center of mass of the hemisphere is
therefore we must lift a weight of water of
rho g (250 pi/3) a distance of (21+15/8) meters
that is in Joules
use rho = 10^3 kg/m^3 and g = 10 m/s^2
- whoops base at 16 meters - Damon, Wednesday, November 10, 2010 at 7:01pm
rho g (250 pi/3) a distance of (16+15/8) meters
- Calculus 2 - Damon, Wednesday, November 10, 2010 at 7:06pm
Now do the cone the same way
base is at 6 meters
volume = (1/3) pi r^2(4)
cg is at [6 - (1/4)4] meters below ground
- Get this now? - Damon, Wednesday, November 10, 2010 at 7:18pm
Can you do the rest now?
- Calculus 2 - Amy, Thursday, November 11, 2010 at 10:30am
Why would you do the distance of hemisphere from 0 to 16, when we are doing just the hemipsher alone then adding it to the rest after.. wouldnt distance by 5-dy
- Calculus 2 - Damon, Thursday, November 11, 2010 at 6:52pm
You are lifting the water from the cg of the hemisphere all the way to the surface.
that is 16 meters to the top of the hemisphere plus another 15/8 to the cg
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