First slide

Fluid statics

Question

Two communicating vessels contain mercury. The diameter of one vessel is n times larger than the diameter of the other. A column of water of height h is poured into the left vessel. The mercury level will rise in the right hand vessel (s = relative density of mercury and $ρ$ = density of water) by

Moderate

A

$\frac{n^{2} h}{{(n + 1)}^{2} s}$
B

$\frac{h}{(n^{2} + 1) s}$
C

$\frac{h}{{(n + 1)}^{2} s}$
D

$\frac{h}{n^{2} s}$

Solution

If the level in narrow tube goes down by $h_{1}$ then in wider tube goes up to $h_{2}$ ,
Now, ${πr}^{2} h_{1} = π {(nr)}^{2} h_{2} \Rightarrow h_{1} = n^{2} h_{2}$
Now, pressure at point A = pressure at point B
$hρg = (h_{1} + h_{2}) ρ^{'} g$
$\Rightarrow h = (n^{2} h_{2} + h_{2}) sg (As s = \frac{ρ^{'}}{ρ})$
$\Rightarrow h_{2} = \frac{h}{(n^{2} + 1) s}$

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