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

Hint: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.