Imagine this, a cylindrical tank with a liquid level height ‘h’. The tank develops a leak in the bottom; Torricelli’s Law says that “Water in an open tank will flow out through a small hole in the bottom with the velocity it would acquire in falling freely from the water level, h.”
If a mass of water ‘m’ fell freely from height ‘h’, the GPE of the mass would be transformed into KE.
mgh=12mv2 [IMG]file:///C:\DOCUME~1\Operator\LOCALS~1\Temp\msohtmlclip1\01 \clip_image002.gif[/IMG]
2gh=v2 [IMG]file:///C:\DOCUME~1\Operator\LOCALS~1\Temp\msohtmlclip1\01 \clip_image004.gif[/IMG]
Therefore according to Torricelli’s Law, the velocity that the water exits the tank is v=2gh, where g = 9.8m/s^{2}.
The change in volume of the tank dVdt=(Venterings)/s-(V(leaving))/s
- Find a relationship for the amount of water leaving the tank per unit time (dVdt [IMG]file:///C:\DOCUME~1\Operator\LOCALS~1\Temp\msohtmlclip1\01 \clip_image002.gif[/IMG], m^{3} per second) using the area of the hole and the velocity of the water. (Volume entering is 0 L/s)
- Use the chain rule dVdt=dhdt×dVdh to develop the differential equation dhdt=f(h) . Solve the equation to find the relationship h(t) [IMG]file:///C:\DOCUME~1\Operator\LOCALS~1\Temp\msohtmlclip1\01 \clip_image008.gif[/IMG].
A full water tank is 1.8m high and has a radius of 2m, a small hole of area 5cm^{2} is created in the bottom of the tank, how long does it take for the tank to empty.
- Consider a tank of the same dimensions as above. The tank is half full when the hole develops and water now enters the tank at a rate of 10L/s. How long does it take to fill/empty the tank?
[IMG]file:///C:\DOCUME~1\Operator\LOCALS~1\Temp\msohtmlclip1\01 \clip_image008.gif[/IMG]