page_title: Torricelli's law
section_title: Clepsydra problem
hierarchical_section_title: Torricelli's law / Clepsydra problem
caption_attribution_description: English: A water clock (clepsydra) Suomi: Vesikello
context_page_description: Torricelli's law, also known as Torricelli's theorem, is a theorem in fluid dynamics relating the speed of fluid flowing from an orifice to the height of fluid above the opening. The law states that the speed v of efflux 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 would acquire in falling freely from a height h, i.e., where g is the acceleration due to gravity. This expression comes from equating the kinetic energy gained, with the potential energy lost, mgh, and solving for v. The law was discovered by the Italian scientist Evangelista Torricelli, in 1643. It was later shown to be a particular case of Bernoulli's principle.
context_section_description: A clepsydra is a clock that measures time by the flow of water. It consists of a pot with a small hole at the bottom through which the water can escape. The amount of escaping water gives the measure of time. As given by the Torricelli's law, the rate of efflux through the hole depends on the height of the water; and as the water level diminishes, the discharge is not uniform. A simple solution is to keep the height of the water constant. This can be attained by letting a constant stream of water flow into the vessel, the overflow of which is allowed to escape from the top, from another hole. Thus having a constant height, the discharging water from the bottom can be collected in another cylindrical vessel with uniform graduation to measure time. This is an inflow clepsydra.
  Alternatively, by carefully selecting the shape of the vessel, the water level in the vessel can be made to decrease at constant rate. By measuring the level of water remaining in the vessel, the time can be measured with uniform graduation. This is an example of outflow clepsydra. Since the water outflow rate is higher when the water level is higher (due to more pressure), the fluid's volume should be more than a simple cylinder when the water level is high. That is, the radius should be larger when the water level is higher. Let the radius  increase with the height of the water level  above the exit hole of area  That is, . We want to find the radius such that the water level has a constant rate of decrease, i.e. .
  At a given water level , the water surface area is . The instantaneous rate of change in water volume is
  From Torricelli's law, the rate of outflow is 
  From these two equations, 
  Thus, the radius of the container should change in proportion to the quartic root of its height,
caption: An inflow clepsydra
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