Torricelli's law.

Question

Does the placement of the water tank affect the value of the height in the formula? For example, the water tank is 5 meters high and it was placed 10 meters above the ground. The leak on the water tank is at the middle. Should I just 2.5 meters for the height or should it be 12.5 meters?

 

Thanks.

Answer 1

I assume you have a tank filled with liquid to the top (station 1), and that it is leaking through a small hole on the side (station 2), and you want to find the speed of the flow v₂ at the leak. 

 

You must reference the top of the tank and the height of the hole to the same reference point.  You can use the height h above the ground, as long as both heights are referenced to ground level.  Then Bernoulli's equation gives 

 

p₁ + ρgh₁ + ρv₁²/2 = p₂ + ρgh₂ + ρv₂²/2

 

p₁ ≅ p₂ ⇒

 

ρgh₁ + ρv₁²/2 = ρgh₂ + ρv₂²/2

 

gh₁ + v₁²/2 = gh₂ + v₂²/2

 

If the horizontal surface area of the the tank is much larger than the area of the hole, then v₁ is negligible compared to v₂, and the continuity equation allows you to use the approximation 

 

gh₁ = gh₂ + v₂²/2 ⇒

 

v₂ = √[2g(h₁ - h₂)]  

 

The is Torricelli's formula.  But note that h₁ - h₂ is the difference in height between the top of the liquid and the hole.  So if the tank is filled to the top, and the hole is in the middle, then h₁ - h₂ is half of the vertical length of the tank, which in this problem is half of 5 meters = 2.5 meters.  That is the same as

 

h₁ - h₂ = (10 + 5)m - (10 + 5/2)m = 2.5 m

 

Does this answer your question?

Answer 2

Yes it does because the formula considers reference point on the ground for height.  So you would have to add the distance from the ground to the bottom of the tank and the height of the tank together.