

Density and Pressure in Fluids
Example 7
Fig 1 
The Utube in Fig 1 contains two liquids in static equilibrium exposed to the atmosphere. Water is in the left arm, and liquid of unknown density is in the right. The water surface is 18 cm above the horizontal line XY and the unknown liquid is 20cm above the same line.
(a)
What is the density of the unknown liquid?
(b) ) If the Utube is used as a manometer to measure gas pressure, what would be the gas pressure if the liquid levels are 10m vertical difference?
Density of water = 1.00 X 10^{3} kgm^{3}
Atmospheric Pressure = 1.01 X 10^{5} Pa

(a) Pressure of the water at A = Atmospheric Pressure + ρ_{w}gh_{w} Eq1
Pressure of the unknown liquid at B = Atmospheric Pressure + ρ_{u}gh_{u} Eq1
Since the liquids are in static equilibrium the pressure is the same
Pressure of the water at A = Pressure of the unknown liquid at B
∴ Atmospheric Pressure + ρ_{w}gh_{w} = Atmospheric Pressure + ρ_{u}gh_{u}
ρ_{w}h_{w} = ρ_{u}h_{u}
ρ_{u}= (ρ_{w}h_{w}) / (h_{u})
ρ_{u}= [(0.18 m X 1.00 X 10^{3} kgm^{3}) (0.20 m) /]
ρ_{u}= 9 X 10^{2} kgm^{3}


Pressure at A = Pressure at B.
Pressure at B = Atmospheric Pressure + ρ_{L}gh_{L}
Pressure at B = (1.01 X 10^{5} Pa) + [(9 X 10^{2} kgm^{3}) X (10 ms^{2} X 10m)]
Pressure at B = 1.91 X 10^{5} Pa

