An inverted conical tank is 3m tall and 1m in diameter at its widest point. The water
is being pumped out of a spout 2m above the top of the tank. Recall that the density
of water is � = 1000kg/m^3
(a) Find the work needed to empty the tank if it is full. Include units.
(b) Find the work to empty half the tank (assuming it is full to begin with).
See the reply to the question above this one from "Jen". Find the mass and m*g, the center of mass, and how far you lift the center of mass.
In all these questions if you lower the outlet with a tight outlet hose you can siphon the water out without doing any work at all :) However to play along assume you lift the water 2 meters above the tank top.
To find the work needed to empty the tank, we can use the concept of gravitational potential energy. The work done is equal to the change in potential energy.
(a) Work needed to empty the tank if it is full:
1. Calculate the initial potential energy when the tank is full. The potential energy of an inverted cone can be calculated using the formula: P.E. = mgh, where m is the mass of the water, g is the acceleration due to gravity, and h is the height from the bottom of the tank to the top.
- The mass of the water can be calculated using its density and volume: m = ρV, where ρ is the density of water and V is the volume of the cone.
- The volume of the cone can be calculated using the formula: V = (1/3)πr²h, where r is the radius of the cone and h is its height.
- In this case, the radius of the widest point of the cone is half of the diameter, so r = 1/2 meter.
- The height of the cone is 3 meters, and the spout is 2 meters above the top, so the total height from the bottom to the top is 3 + 2 = 5 meters.
- Plug in the values into the formula and calculate the initial potential energy.
2. Calculate the final potential energy when the tank is empty. The final height is 2 meters above the top of the tank.
- Plug in the values into the formula and calculate the final potential energy.
3. Subtract the final potential energy from the initial potential energy to find the work needed to empty the tank.
(b) Work to empty half the tank (assuming it is full):
1. Calculate the initial potential energy when the tank is full. The formula and calculations are the same as in part (a).
2. Calculate the final potential energy when the tank is half full. Since we are only emptying half the tank, the final height will be halfway between the top and bottom (2.5 meters above the bottom).
- Plug in the values into the formula and calculate the final potential energy.
3. Subtract the final potential energy from the initial potential energy to find the work needed to empty half the tank.
Note: In both cases, make sure to convert all units to the appropriate system (e.g. meters, kilograms, etc.), and use the correct value for the acceleration due to gravity (usually 9.8 m/s²).