The Manometer

Figure illustrates a manometer consisting of a U-tube, the two arms of which are associated to two systems in which the pressures are p1 and p2. The aim of the manometer is to give a measure of the difference among p₁ and p₂. The lower part of the U-tube is filled with a liquid having density that greatly exceeds the densities of the fluids in the two systems. The vertical distance z dividing the two free liquid surfaces is measured by means of a properly supported scale.  

From the principles of hydrostatics, the density, pressure and height difference are associated by:

                                                                       (p₁ – p₂) = ρ gz                                            

Knowing ρ, measurement of z allows the pressure difference (p₁ − p₂) to be computed. Additionally, when p₂ is known, p₁ can be established.
It is fascinating to note that the two arms of the U-tube require not be of equivalent, or even of uniform cross-sectional area.

Example:

Compute the pressure difference in Pa equivalent to 1 mm difference in liquid surface level in a manometer if the liquid is: (i) water, (ii) mercury, acquire the densities of water and mercury to be 1000 and 13568 kg/m3; and the gravitational acceleration to be 9.81 m/s2.

Solution:

                                                                     (p₁ – p₂) = ρ gz

(a)  For water, (p₁ – p₂) = 1000 kg/m³ × 0.001 m × 9.81 m/s² = 9.81 Pa

(b)  For mercury, (p₁ – p₂) = 13568 kg/m³ × 0.001 m × 9.81 m/s² = 133.1 Pa