First slide

Pressure in a fluid-mechanical properties of fluids

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 h₁, then in wider tube the level goes up to h₂,
$Now, π r^{2} h_{1} = π (n r)^{2} h_{2} \Rightarrow h_{1} = n^{2} h_{2}$
Now, pressure at point A = pressure at point B
$h ρ g = (h_{1} + h_{2}) ρ^{'} g$
$\Rightarrow h = (n^{2} h_{2} + h_{2}) s g (As s = \frac{ρ^{'}}{ρ}) \Rightarrow h_{2} = \frac{h}{(n^{2} + 1) s}$

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