A tank in the shape of an inverted right circular cone has height 5 meters and radius 3 meters. It is filled with 3 meters of hot chocolate. Find the work required to empty the tank by pumping the hot chocolate over the top of the tank. The density of hot chocolate is 1100kg/m^3 Your answer must include the correct units
When h=3, r = 9/5 so the volume of liquid is
1/3 πr^2 h = 1/3 π (9/5)^2 * 3 = 81π/25 m^3
so the weight is 1100 * 9.81 * 81π/25 N
The center of gravity of the liquid is at height 3/4, so the weight must be lifted through a height of 4.25 m
That makes the work 1100 * 9.81 * 81π/25 * 4.25 = 466,816 J
If no one comes by while I'm away, to post the calculus way to solve this, I'll take care of that later. But just think of the work involved in lifting a disc of thickness dh through a distance of 5-h.
To find the work required to empty the tank, we need to calculate the gravitational potential energy of the hot chocolate in the tank.
The formula to calculate gravitational potential energy is given by:
Potential Energy = Mass x Gravity x Height
To find the mass of the hot chocolate, we need to calculate the volume of the hot chocolate and multiply it by its density:
Volume of hot chocolate = Volume of cone - Volume of empty space above the hot chocolate
= (1/3)πr^2h - (1/3)πr^2(5)
= (1/3)π(3^2)(3) - (1/3)π(3^2)(5)
= (1/3)π(9)(3) - (1/3)π(9)(5)
= 9π - 15π
= -6π
Note: The negative volume indicates that there is empty space above the hot chocolate.
Now, we can calculate the mass of the hot chocolate:
Mass of hot chocolate = Volume of hot chocolate x Density
= -6π x 1100 kg/m^3
= -6600π kg
Next, we'll calculate the height of the hot chocolate.
The height of the hot chocolate is given as 3 meters, which is also the height of the filled part of the tank. Since the tank is inverted, the height of the hot chocolate is the difference between the height of the tank and the height of the filled part:
Height of hot chocolate = Height of tank - Height of filled part
= 5 - 3
= 2 meters
Now, we can calculate the gravitational potential energy of the hot chocolate:
Potential Energy = Mass x Gravity x Height
= (-6600π kg) x (9.8 m/s^2) x (2 m)
= -128520π J
The work required to empty the tank by pumping the hot chocolate over the top is equal to the negative of the potential energy:
Work required = -Potential Energy
= 128520π J
Therefore, the work required to empty the tank by pumping the hot chocolate over the top is 128520π joules.