Question

A barometer is faulty. When the true barometer reading are 73 and 75 cm of Hg, the faulty barometer reads 69 cm and 70 cm respectively. What is the true reading when the faulty barometer reads 69.5 cm?

Answer 1

Let the density of mercury and cross section area of barometer be $\rho$ and $A$ respectively.

For faulty barometer, $P_o=\rho gh+P_{air}$

Case (a) : $\rho g\left(73\right)=\rho g\left(69\right)+P_1$ $⟹P_1=\rho g\left(4\right)$

Case (b) : $\rho g\left(75\right)=\rho g\left(70\right)+P_2$ $⟹P_2=\rho g\left(5\right)$

Also $P_1{V}_1=P_2{V}_2$

$\therefore$ $\rho g\left(4\right)\times Ax=\rho g\left(5\right)\times A(x-1)$ $⟹x=5cm$

Case (c) : Let the true pressure be shown by barometer reading $h$

Using $P_1{V}_1=P_3{V}_3$ where $V_3=A(x-0.5)=A\left(4.5\right)$

$\therefore$ $\rho g\left(4\right)\times A\left(5\right)=P_3\times A\left(4.5\right)$ $⟹P_3=\rho g\left(4.44\right)$

$\therefore$ $\rho gh=\rho g\left(69.5\right)+\rho g\left(4.44\right)$ $⟹h=73.94cm$

The true reading of the barometer will be 73.94 cm.