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!
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:
http://mathworld.wolfram.com/Hemisphere.html
Therefore the center of mass of the hemisphere is
21 +(3/8)5
below earth
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
rho g (250 pi/3) a distance of (16+15/8) meters
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
Can you do the rest now?
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
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
To find the work required to empty the tank, we need to calculate the work done for each part of the tank separately and then sum them up. Let's start by finding the work for each section.
1. Hemisphere:
The work done to lift the water from the hemisphere can be calculated as the product of the force required and the distance the water is lifted. Since the water is lifted vertically against gravity, the force required is given by the weight of the water.
To find the weight of the water in the hemisphere, we first need to calculate the volume of the hemisphere. The volume V_hemi of a hemisphere can be calculated using the formula V_hemi = (2/3)πr^3. Substituting the radius r = 5 m into the formula, we find V_hemi = (2/3)π(5^3) = (2/3)π(125) = 250π/3 m^3.
The weight W_hemi of the water in the hemisphere can be calculated using the formula W_hemi = density × volume × gravity. Substituting the given values, we find W_hemi = 1000 kg/m^3 × (250π/3) m^3 × 10 m/s^2.
Now, to calculate the work done to lift the water out of the hemisphere, we multiply the weight of the water by the distance it is lifted. Since the top of the tank is 2 m below the ground surface, the water needs to be lifted by a distance of 2 m.
Therefore, the work done to empty the hemisphere is given by W_hemi = W_hemi × 2.
2. Cylinder:
Similar to the hemisphere, we calculate the volume V_cyl of the cylinder using the formula V_cyl = πr^2h. Substituting the values, we find V_cyl = π(5^2)(10) = 250π m^3.
The weight W_cyl of the water in the cylinder is given by W_cyl = density × volume × gravity, which becomes W_cyl = 1000 kg/m^3 × (250π) m^3 × 10 m/s^2.
The water needs to be lifted by a distance of 10 m (the height of the cylinder), so the work done to empty the cylinder is given by W_cyl = W_cyl × 10.
3. Circular cone:
Similarly, we calculate the volume V_cone of the cone using the formula V_cone = (1/3)πr^2h. Substituting the values, we find V_cone = (1/3)π(5^2)(4) = 100π/3 m^3.
The weight W_cone of the water in the cone is given by W_cone = density × volume × gravity, which becomes W_cone = 1000 kg/m^3 × (100π/3) m^3 × 10 m/s^2.
Since the water in the cone needs to be lifted by a distance of 14 m (the height of the cylinder plus the height of the cone), the work done to empty the cone is given by W_cone = W_cone × 14.
Finally, to find the total work required to empty the tank, we add up the work done for each section:
Total Work = Work_hemi + Work_cyl + Work_cone.
Please substitute the calculated values and follow the steps outlined to find the final answer for the work required to empty the tank.