The coefficient of cubical expansion of mercury is $0.00018{/}^{°}C$ and that of brass $0.00006{/}^{°}C$. If a barometer having a brass-scale were to read $74.5cm$ at $30{}^{°}C$, find the true barometric height (in cm) at $0{}^{°}C$. The scale is supposed to be correct at ${\text{15\hspace{0.17em}}}^{°}C$(round off to nearest integer)

Coefficient of linear expansion of brass,

                                                $\alpha =\frac{\gamma }{3}=\frac{0.00006}{3}$

                                                $=0.00002{/}^{°}C$

The brass scale is true at $15{}^{°}C$, therefore at ${\text{30\hspace{0.17em}}}^{°}C$ its graduation will increase in length and so observed reading will be less than true reading by,

$\mathrm{\Delta }\ell =\ell \alpha \mathrm{\Delta }t=74.5\times 0.00002\times 15$

$=0.02235cm$

$\therefore$   True reading at $30{}^{°}C$,

${\mathcal{l}}_{30}={\mathcal{l}}_{observed}+\Delta \mathcal{l}$

$\begin{array}{l}=74.5+0.02235\\ =74.522cm\end{array}$

As pressure is constant at all temperature, so Pressure at $0^{°}C$ $=$Pressure  at $30{}^{°}C$

or

$h_0{\rho }_{0}g=h_{30}{\rho }_{30}g$

$h_0=\frac{h_{30}{\rho }_{30}}{{\rho }_0}=\frac{h_{30}}{{\rho }_0}\times \frac{{\rho }_0}{\left(1+\gamma H{g}^t\right)}$

$=\frac{74.522}{1+0.00018\times 30}=74.122cm$