Question

A cylindrical vessel open at the top is $20cm$ high and $10cm$ in diameter. A circular hole whose cross-sectional area $1{cm}^2$ is cut at the centre of the bottom of the vessel. Water flows from a tube above it into the vessel at the rate $100{cm}^3{s}^{-1}$. The height of water in the vessel under steady state (in cm) is

Answer 1

When the steady state is reached, rate of outflow of water will be equal to inflow of water.
$\Rightarrow$Rate of outflow = 100 cm${}^3$ $\because$ Rate of inflow = 100 cm${}^3$
$\Rightarrow av=100$ cm${}^3$, where $a$ is the cross-section area of circular
hole and $v$ is the velocity of water coming out of the hole.

$\Rightarrow a\sqrt{2gh}=100$ cm${}^3\because v=\sqrt{2gh}$ using Torricelli's theorem.

$\Rightarrow 1\times \sqrt{2gh}=100$ cm${}^3$

$\Rightarrow \sqrt{2gh}=100$ cm${}^3$

$2gh=10000\Rightarrow 2\times 1000\times h=10000\Rightarrow h=5cm$