An underground tank full of water has the shape of a circular cone with a base radius 5m and height 4m at the top. The top of the tankl is 2m 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 to the surface. Density= 1000kg/m^3 and gravity= 10 m/s^2)
To find the work required to empty the tank, we need to calculate the potential energy of the water when it is pumped out to the surface.
First, let's determine the volume of water in the tank. The volume of a cone can be calculated using the formula:
V = (1/3) * π * r^2 * h
where V is the volume, π is a mathematical constant approximately equal to 3.14159, r is the base radius, and h is the height of the cone.
Given that the base radius (r) is 5m and the height (h) is 4m, we can substitute these values into the formula to calculate the volume (V):
V = (1/3) * π * 5^2 * 4
= (1/3) * π * 25 * 4
≈ 104.67 m^3
Next, we can calculate the mass of the water in the tank by multiplying the volume by the density. Given that the density is 1000 kg/m^3, we have:
Mass = Density * Volume
= 1000 * 104.67
= 104670 kg
To find the potential energy, we need to multiply the mass of the water by the height it is raised. In this case, the height is the distance from the top of the tank to the surface, which is 2m. We will use the gravitational potential energy formula:
Potential Energy = Mass * Gravity * Height
= 104670 * 10 * 2
= 2093400 J
Therefore, the work required to empty the tank by pumping all of the water out to the surface is approximately 2,093,400 Joules.