Question

The height of a mercury barometer is $75cm$ at sea level and $50cm$ at the top of a hill a . Ratio of density of mercury to that of air is ${10^4}$. The height of the hill is-
A. $1.25Km$
B. $2.5Km$
C. $250m$
D. $750m$

Answer 1

A mercury barometer is a device that is used to measure the atmospheric pressure at a given location. As, ratio of density of mercury to that of the air, $\dfrac{{{\rho _{Hg}}}}{{{\rho _{Air}}}}$ is given=${10^4}$. We know the equation for the change in pressure. By substituting all the given values, we can easily find the value of the height, h.

Formula used:
$\Delta p = \left( {{h_1} - {h_2}} \right) \times {\rho _{Hg}} \times g$
Here $\Delta p$ is the change in pressure,${h_1}$ and ${h_2}$ are the heights of
barometer, g is gravity and ${\rho _{Hg}}$ is the density of mercury.

Complete step by step answer:
As we know that the pressure difference between the sea level and the top of hill is-
$\Delta p = \left( {{h_1} - {h_2}} \right) \times {\rho _{Hg}} \times g$ ---- (1)
${h_1}$ and ${h_2}$ are the heights of mercury barometer- given- $75cm$ and $50cm$ respectively. Now substitute all the values in the equation (1), we get-
$\Delta p = \left( {{h_1} - {h_2}} \right) \times {\rho _{Hg}} \times g$
$\Rightarrow\Delta p = \left( {75 - 50} \right) \times {10^{ - 2}} \times {\rho _{Hg}} \times g$ --- (2)
Pressure difference due to h metre of air-$\Delta p = h \times {\rho _{Air}} \times g$-- (3)

Equate equation (2) and (3), we get-
$h \times {\rho _{Air}} \times g$=$\left( {75 - 50} \right) \times {10^{ - 2}} \times {\rho _{Hg}} \times g$
For finding the height of the hill, h we can take all terms on the right hand side, we get-
$\dfrac{{{\rho _{Hg}}}}{{{\rho _{Air}}}} \times 25 \times {10^{ - 2}}$
Now we know the ratio of density of mercury to the air is already given in this question,
$\therefore h = {10^4} \times 25 \times {10^{ - 2}}$
So, the height of the hill comes out to be $2500m$or $2.5Km$ .

Hence, option B is correct.

Note: A mercury barometer is a device that is used to measure the atmospheric pressure at a given location. The barometer consists of a vertical glass tube which is closed at one end. Additionally, The air around us has weight, and it presses against everything it touches. That pressure is known as atmospheric pressure.