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A large cylindrical tank has a hole of area \( A \) at its bottom and water is poured into the tank through a tube of cross-sectional area \( \boldsymbol{A} \) ejecting water at the speed \( v \). Which of the following is true?

1. Water level in tank keeps on rising.
2. No water can be stored in the tank
3. water level will rise to a height \( \frac{v^{2}}{2 g} \) and then stop.
4. The water level will be oscillating.

Correct answer: water level will rise to a height \( \frac{v^{2}}{2 g} \) and then stop.

Solution

The water level rises until the hydrostatic pressure at the hole makes the outflow speed equal the inflow speed. Setting Torricelli’s law \(\sqrt{2gh}=v\) gives \(h=\frac{v^2}{2g}\), so after that the inflow and outflow balance. Therefore the level stops rising at that height.

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