A large cylindrical tank has a hole of area A at its bottom....

A large cylindrical tank has a hole of area $A$ at its bottom. Water is poured in the tank by a tube of equal cross-sectional area $A$ ejecting water at the speed $v$. (A) The water level in the tank will keep on rising. (B) No water can be stored in the tank. (C) The water level will rise to a height $v^{2} / 2 g$ and then stop. (D) The water level will oscillate.

A large cylindrical tank has a hole of area $A$ at its bottom. Water is poured in the tank by a tube of equal cross-sectional area $A$ ejecting water at the speed $v$.
(A) The water level in the tank will keep on rising.
(B) No water can be stored in the tank.
(C) The water level will rise to a height $v^{2} / 2 g$ and then stop.
(D) The water level will oscillate.

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Key Concepts

Torricelli's Law

This principle describes how the speed of a fluid exiting a hole under gravity is determined by the height of the fluid above the hole. It shows that the exit speed is proportional to the square root of the height, and it is crucial in understanding how the outflow rate changes as the water level varies in the tank.

Continuity Equation

This concept is based on the conservation of mass in fluid dynamics. It states that, for an incompressible fluid, the rate of flow into a system must equal the rate of flow out unless the fluid is being stored. In this problem, the net accumulation of water in the tank depends on the balance between the constant inflow rate and the gravity-driven outflow rate.

Dynamic Equilibrium

A dynamic equilibrium in a fluid system occurs when the inflow and outflow rates are equal. In the context of this problem, the water level rises until the outflow speed, given by Torricelli's law, reaches the inflow speed. At that point, the system reaches an equilibrium state with a constant water level.

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