Question

The height of mercury barometer is h when the atmospheric pressure is ${10}^{5}Pa$, the pressure at x in the shown figure is

A. ${10}^{5}Pa$
B. $0.8\times {10}^{5}Pa$
C. $0.2\times {10}^{5}Pa$
D. $120\times {10}^{5}Pa$

Answer 1

Hint: The above figure shows a setup of a static fluid. The horizontal surface of mercury will experience an exertion by the atmosphere. Pressure at point y can be calculated by formula of pressure. So by equating two equations for pressure at the horizontal surface we can easily encounter our answer.
As per the given data;
Height at which point x is situated $=h-{\displaystyle \frac{h}{5}}$
Atmospheric pressure $P_{\circ }={10}^5$ Pa

Formula used:
Pressure, $P=P_{\circ }+\rho gh$

Complete answer:
Hydrostatic pressure refers to the pressure exerted by a fluid (gas or liquid) at any point in space within that fluid, assuming that the fluid is incompressible and at rest.
Pressure within a liquid depends only on the density of the liquid, the acceleration due to gravity, and the depth within the liquid. The pressure exerted by such a static liquid increases linearly with increasing depth.

Mathematically;
Pressure, $P=P_{\circ }+\rho gh$
Where, $P_{\circ }$ the pressure due to empty space or atmospheric formula
So pressure at the bottom surface of the glass tube of the barometer $(P_y)$ .will be;
(Assume that pressure due to the empty space in the glass tube is zero)
$$P_y=0+\rho gh$$
$$P_y=\rho gh$$
Let us see the pressure at y due to point x under the glass tube:
$$P_x=\rho gh-\rho g{\displaystyle \frac{h}{5}}$$
 $P_x=\rho gh-\rho g{\displaystyle \frac{h}{5}}$
$$=\rho gh(1-{\displaystyle \frac{1}{5}})$$
We know that, In case of static fluid, pressure on the horizontal level is constant at every point.
Thus;
$$P_y=\rho gh={10}^{5}Pa$$
By putting the value of$\rho gh$:
$P_x={10}^5(1-{\displaystyle \frac{1}{5}})$
$$={10}^5\times {\displaystyle \frac{4}{5}}$$
$$P_x=0.2\times {10}^5$$

So, the correct answer is “Option C”.

Note:
Read the question carefully. Remember that we don’t have to consider the whole height of the glass tube. Most importantly this is a static fluid setup so apply properties according to that only.