Torricelli's theorem

Japanese: トリチェリの定理 - とりちぇりのていり

Put liquid in a container and make a hole in the wall of the container. If the hole is small enough compared to the size of the container, the flow can be considered to be in a steady state. In this case, the speed v of the liquid flowing out of the hole is

This is called Torricelli's theorem ( see figure ). Here, g is the acceleration due to gravity, 9.8 m/s ² , and h is the depth from the water surface to the hole. Torricelli's theorem can be proven using Bernoulli's theorem: P + (1/2)ρ v ² + ρg z = constant (ρ is the density of the liquid). If the position of the hole is taken as z = 0, the height of the water surface is at z = h , and since the container is large enough, we can assume that the speed at which the water surface falls is zero. The position of the hole and the pressure on the water surface are both equal to the atmospheric pressure p _∞ . Therefore,
P _∞ +ρg h = P _∞ +(1/2)ρ v ²
holds true, and the velocity of the liquid flowing out of the hole is

It is required.

[Ryo Ikeuchi]

[References] | Torricelli | Bernoulli's theorem

Torricelli's theorem (diagram)
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Torricelli's theorem (diagram)

Source: Shogakukan Encyclopedia Nipponica About Encyclopedia Nipponica Information | Legend

Japanese:

器に液体を入れ、器の壁に穴をあける。穴は容器の大きさに比べ十分小さいとすると、流れは定常状態としてよい。このとき穴から流れ出す液体の速さvは、

で表される。これをトリチェリの定理という（図）。ここでgは重力加速度9.8m/s²、hは水面から穴までの深さである。トリチェリの定理はベルヌーイの定理
　　P＋(1/2)ρv²＋ρgz＝一定
を用いると証明できる（ρは液体の密度）。穴の位置をz＝0にとると、水面の高さはz＝hの位置にあり、容器が十分大きいため水面の下がる速度はゼロと置ける。穴の位置および水面にかかる圧力はともに大気圧p_∞に等しい。したがって、
　　P_∞＋ρgh＝P_∞＋(1/2)ρv²
の等式が成り立ち、穴から流れ出す液体の速度が

と求められる。