Question

The height of a mercury barometer is 75cm at sea level and 50cm at the top of a hill. Ratio of density of mercury to that of air is 104. The hill is:
A) 1.25Km
B) 2.5Km
C) 250m
D) 750m

Answer 1

The above problem is based on the concept of pressure exerted by liquid when the liquid is at rest which is equal to the height of the liquid density of the liquid and gravitational acceleration.
$P = \rho gh$ ($\rho$is the density of the liquid, h is the height of the liquid and g is the acceleration).
Using the above mentioned relation we will find the height of the hill.

Complete step by step solution:
Let us discuss some of the properties of the pressure exerted by the liquid kept in a vessel.
When a liquid of some density is kept in a vessel with height h. The weight of the liquid column exerts a downward force and thus downward pressure and hence we have the formula:
$P = \rho gh$
Let us do the calculation now:
Pressure difference caused due to sea level and top of the hill;
$\Rightarrow \Delta P = ({h_1} - {h_2}){\rho _m}g$ (Height at sea level and hill top is subtracted and ${\rho _m}$ is density of mercury)
$\Rightarrow \Delta P = (75 - 50){\rho _m}g \\\ \Rightarrow \Delta P = 25{\rho _m}g \\\$(We substituted the numerical value of height)..................(1)
Pressure difference due to air with height h
$\Rightarrow \Delta P = {\rho _a}hg$ ..................(2)
Equating equation 1 and 2
$\Rightarrow {\rho _a}hg = 25{\rho _m}hg \\\ \Rightarrow h = 25 \times {10^4} \\\$(Ratio of density of mercury to air is 104)
The above calculated quantity is in cm to convert it in m we will multiply by $10^{-2}$
$\Rightarrow h = 25 \times {10^4} \times {10^{ - 2}} \\\ \Rightarrow h = 2.5Km \\\$
Note: We will have many other applications where pressure has different effects for example a sharp knife cuts easily while blunt knife take much force in cutting, when we stand on our feet pressure is more than we lay down, sharp needle is easily able to pierce the skin rather than a dull needle, camel can easily walk on sand than human beings because area of feet of camel is larger than human being.