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
There are at least two ways to do this.
You can calculate the distance below the ground of the center of gravity of the water in the tank. You are then lifting the mass of water from that point to the ground and work = m g times that distance up.
The other way is to find the cross sectional area of every slice of the tank at distance below ground z, call it Az. The work done on a slice is then rho g Az dz . Integrate that from z = 0 to z = 16.
The work done on a slice is then rho g Az z dz . Integrate that from z = 0 to z = 16.
(forgot distance slice is lifted, z)
To find the work required to empty the tank, we need to calculate the potential energy of the water in the tank. The potential energy is defined as the work done against gravity to lift the water from its initial position to the final position.
First, let's break down the tank into its different parts: the hemisphere, the cylinder, and the cone.
1. Hemisphere:
The volume of a hemisphere is given by V = (2/3) * π * r³, where r is the radius. In this case, the radius is 5 m. Let's calculate the volume:
V_hemi = (2/3) * π * 5³ = (2/3) * 3.14 * 125 = 261.67 m³
The height of the hemisphere is not needed in this calculation since we are considering the entire volume.
2. Cylinder:
The volume of a cylinder is given by V = π * r² * h, where r is the radius and h is the height. In this case, r = 5 m and h = 10 m. Let's calculate the volume:
V_cyl = π * 5² * 10 = 250π = 785.4 m³
3. Cone:
The volume of a cone is given by V = (1/3) * π * r² * h, where r is the radius and h is the height. In this case, r = 5 m and h = 4 m. Let's calculate the volume:
V_cone = (1/3) * π * 5² * 4 = 83.33π = 261.67 m³
Now let's calculate the total volume of water in the tank:
Total volume = V_hemi + V_cyl + V_cone = 261.67 + 785.4 + 261.67 = 1308.74 m³
Next, we need to convert the volume of water to the mass of water using the density of water:
Mass = density * volume = 1000 kg/m³ * 1308.74 m³ = 1,308,740 kg
Now, we can calculate the work required to lift the water from the bottom of the tank to the surface. The work is given by the formula:
Work = Force * Distance
The force is equal to the weight of the water, which can be calculated using:
Force = Mass * Gravity = 1,308,740 kg * 10 m/s² = 13,087,400 N
The distance is the height from the top of the tank to the surface, which is given as 2 m.
Now, we can calculate the work required:
Work = Force * Distance = 13,087,400 N * 2 m = 26,174,800 Nm or 26,174,800 J (Joules)
Therefore, the work required to empty the tank by pumping all of the water out is 26,174,800 Joules.