Applications on pascal’s principle , Manometer types and uses
The aqueous manometer is preferred to the mercuric one to measure small pressure differences because the density of water is small compared to that of mercury , So , the difference in water levels in the two branches of the manometer becomes clear and easy to be measured which decreases the error in measurement .
The manometer consists of U-shaped tube containing proper amount of liquid of known density , One of its ends is connected to the gas reservoir whose pressure to be measured and the other end exposed to air , The pressure at all points in the same horizontal plane in a homogeneous liquid is the same .
Manometer is used in measuring the pressure of enclosed gas , It is used in measuring the difference between the pressure of enclosed gas and the atmospheric pressure .
- Aqueous manometer : the liquid used is water .
- Mercuric manometer : the liquid used is mercury .
It is preferable to use the mercuric manometer for measuring high pressure difference because the mercury has high density , so , mercury neither rash out the tube nor into the gas reservoir .
How it is used ?
If the liquid level in the free end branch is the same liquid level in the other branch that connected to the gas reservoir , Then , Pa = Pgas
Δ P = Pgas − Pa = zero
Δ P = zero , If the liquid used is mercury
If the liquid level in the free end branch is higher than the liquid level in the other branch that connected to the gas reservoir , Then , Pgas > Pa .
Pgas = Pa + ρ g h
Δ P = Pgas − Pa
Δ P = + ρ g h ( N / m² )
If the liquid used is mercury then : Pgas = Pa + h
Δ P = Pgas − Pa , Δ P = + h ( cm Hg ) .
If the liquid level in the free end branch is lower than the liquid level in the branch that connected to the gas reservoir , Then , Pgas < Pa .
Pgas = Pa − ρ g h
Δ P = Pgas − Pa
Δ P = − ρ g h ( N / m² )
If the liquid used is mercury then : Pgas = Pa − h
Δ P = Pgas − Pa , Δ P = − h ( cm Hg ) .
When a liquid is put in a glass container equipped with a piston at the top , then the pressure at point A at depth h is :
P = P1 + ρ g h
Where : P1 is the pressure at the liquid surface and it is due to the atmospheric pressure and the weight of the piston , ρ g h is the pressure of the liquid column on point A , when putting extra load on the piston , the pressure increases by Δ P .
P = P1 + ρ g h + Δ P
The piston does not move down because the liquid is incompressible , If the pressure on the piston is increased to a certain limit , the glass container breaks down , This means that the pressure acting on the piston is transferred to all parts of the liquid and to the walls of the container , The French scientist Pascal set his principle based on this result .
Pascal’s principle : When pressure is applied on a liquid in a container , the pressure is transmitted in full to all parts of the liquid as well as the walls of the container , Pascal’s principle is not applied to gases because unlike liquids , they can be compressed due to the big intermolecular spaces between their molecules .
Applications on pascal’s principle
- The hydraulic press
- The hydraulic brakes of the car
- The hydraulic lift ( It uses a liquid to lift cars in the petrol stations )
- Chair of the dentist
- The hydraulic drill
- Diving suit
The hydraulic press
Hydraulic press consists of a tube with 2 pistons at its ends , one small of area ( a ) and the other big of area ( A ) where the space between them is filled with an appropriate liquid such as oil , It is used in lifting high loads using a small force and it is based on Pascal’s principle .
When a force ( f ) acts on the small piston , pressure ( P ) is produced , P = f / a , The pressure ( P ) transfers completely through the liquid to the lower surface of the big piston where a force ( F ) is produced , P = F / A , In this case the pistons are in equilibrium at one horizontal level where :
Pressure on the small piston = Pressure on the big piston
P = f / a = F / A
If the force ( f ) moved the small piston a distance ( y1 ) , then the big piston is affected by a force ( F ) that moves it a distance ( y2 ) .
Applying the law of conservation of energy ( case of ideal press ) , then :
Work done on the small piston = Work done on the big piston
f y1 = F y2
F/ f = y1 / y2
Cases of hydraulic press
When the two pistons are at the same level
P = f / a = F / A
When the two pistons are at different levels
P = ( f / a ) = ( F / A ) + ρ g h or P = ( f / a ) + ρ g h = F / A
Where : ρ is the liquid density , h is difference in height between the two pistons .
The mechanic advantage of the hydraulic press
The mechanic advantage ( η ) is determined from the relation :
η =( F / A ) = ( f / a ) = ( A / a ) = ( R² / r² ) = ( D² / d² ) = ( y1 / y2 )= ( v1 / v2 )
Where : R = radius of the big piston , r = radius of the small piston
D = diameter of the big piston , d = diameter of the small piston
v1 = speed of the small piston , v2 = speed of the big piston
The mechanic advantage ( η ) is the ratio between the force produced at the big piston and the acting force at the small piston or it is the ratio between the cross-section area of the big piston to the cross-section area of the small piston .
When the mechanical advantage of a hydraulic press at equilibrium = 400 , It means that the ratio between the force produced at the big piston to that acting at the small piston = 400 .
Efficiency of the hydraulic press is the ratio between the work done at the big piston to the work done at the small piston , The efficiency of the hydraulic press is determined from the relation : F y2 ÷ f y1
When the efficiency of the hydraulic press = 97% , It means that the ratio between the work done at the big piston to the work done at the small piston = 97 / 100 .
The efficiency of the hydraulic press does not reach 100 % because the friction between the piston and the walls of the tube , the gas bubbles in the liquid where work is consumed to reduce the volume of bubbles .
The hydraulic brake system in a car is two types
Drum brake ( rear brake ) : It uses Pascal’s rule as the braking system uses a brake fluid , Upon pushing on the brake pedal with a small force and a relatively long stroke ( distance ) , the pressure is transmitted in the master brake cylinder , hence , onto the liquid and the whole hydraulic line , then to the piston of the wheel cylinder outwardly and finally to the brake shoes and the brake drum , A force of friction results in , which eventually stops the car .
Front ( disk ) brake : In this case , the forces resulting from the braking action press on the brake pads which produce friction enough to stop the wheel , It should be noted that the distance travelled by the brake shoes in both cases is small because the force is large .
It protects the diver from the high pressure in the very deep seas where air is blown inside it to protect the head of the driver from the high pressure .
Applications on the pressure at a point ( Connected vessels , U-shaped tube & Mercuric barometer )
Properties of fluids , Factors affecting density and pressure
Steady flow , Turbulent flow and Applications on the continuity equation