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 $\rho =$ density of water) by

• A. $\frac{n^{2} h}{(n+1)^{2} s}$
• B. $\frac{h}{\left(\right. n^{2} + 1 \left.\right) \, s}$
• C. $\frac{h}{(n+1)^{2} s}$
• D. $\frac{h}{n^{2} s}$

Answer 1

Answer: (B)

If the level in narrow tube goes down by $h_1$ then in wider tube goes up to $h_2$
Now, $\pi r^{2} h_{1}=\pi(n r)^{2} h_{2}$
$\Rightarrow h_{1}=n^{2}h_{2}$
Now, pressure at point $ A = $ pressure at point $ B$
$h\rho g=\left(\right.h_{1}+h_{2}\left.\right)\rho'g$
$\Rightarrow h \, =\left(n^{2} h_{2} + h_{2}\right)sg\left(A s \, s = \frac{\left(\rho \right)'}{\rho }\right)$
$\Rightarrow h_{2}=\frac{h}{\left(\right. n^{2} + 1 \left.\right) s}$