Torricelli's law

Not to be confused with Torricelli's (Roger's) equation or Torricelli point.

Torricelli's (Roger's) law, also known as Torricelli's theorem, is a theorem in fluid dynamics relating the speed of fluid flowing out of an opening to the height of fluid above the opening.

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Torricelli's law states that the speed of efflux, v, of a fluid through a sharp-edged hole at the bottom of a tank filled to a depth h is the same as the speed that a body (in this case a drop of water) would acquire in falling freely from a height h, i.e. \(v = \sqrt{2gh}\), where g is the acceleration due to gravity. This last expression comes from equating the kinetic energy gained, \(\frac{1}{2}mv^2\), with the potential energy lost, mgh, and solving for v.

The law was discovered (though not in this form) by the Italian scientist Evangelista Torricelli, in 1643. It was later shown to be a particular case of Bernoulli's principle.

Derivation

Bernoulli's principle states that:

\[{v^2 \over 2}+gz+{p\over\rho}=\text{constant}\]

where v is fluid speed, g is the gravitational acceleration, z is the fluid's height above a reference point, p is pressure, and ρ is density. Define the opening to be at z=Ø. At the top of the tank, p is equal to the atmospheric pressure. v can be considered 0 because the fluid surface drops in height extremely slowly compared to the speed at which fluid exits the tank. At the opening, z=Ø and p is again atmospheric pressure. Eliminating the constant and solving gives:

\[gz+{p_{atm}\over\rho}={v^2 \over 2}+{p_{atm}\over\rho}\] \[\Rightarrow v^2=2gz\,\] \[\Rightarrow v=\sqrt{2gz}\]

z is equivalent to the h in the first paragraph of this article, so:

\[v=\sqrt{2gh}\]

References

• Stanley Middleman, An Introduction to Fluid Dynamics: Principles of Analysis and Design (John Wiley & Sons, 1997) ISBN 978-0-471-18209-2
• Dennis G. Zill, A First Course in Differential Equations (2005)

See also

• Spouting can experiment
• Pascal's law

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