1. pressure

Introduction

Liquids and gases can flow and are therefore called fluids. It is this property that distinguishes liquids and gases from solids in a basic way. Fluids are everywhere around us. Earth has an envelope of air and two-thirds of its surface is covered with water. Water is not only necessary for our existence; every mammalian body consists mostly of water. All the processes occurring in living beings including plants are mediated by fluids. Thus understanding the behaviour and properties of fluids is important.

How are fluids different from solids? What is common in liquids and gases?

Solids have fixed volume and shape, liquids have fixed volume but not fixed shape. Gases on the other hand neither have fixed volume nor fixed shape.

Fluids can be defined as any substance which is capable of flowing.

They don’t have any shape of their own. For example:-water which does not have its own shape but it takes the shape of the container in which it is poured. But when we pour water in a tumbler it takes the shape of the tumbler

Fluids are assumed to be the incompressible (i.e., the density of liquid is not dependent on the variation in pressure and remains constant).

Fluids are also assumed to be non-viscous (i.e., the two liquid surfaces in contact are not pressing any tangential force on each other)

Pressure

Pressure is defined as the physical force exerted on an object. The force applied is perpendicular to surface of objects per unit area.

Mathematically ,  P= F/ A

Unit of pressure is Pascals (Pa).

Since P= F/A   so pressure is inversely proportional to the Area.  If the area would be less, pressure would be more.

For Example:-

Consider a very sharp needle which has a small surface area and consider a pencil whose back is very blunt and has more surface area than the needle.

If we poke a needle in our palm it will hurt as the needle gets pierced inside our skin. Whereas if we poke the blunt side of the pencil into our hand it won’t hurt so much. This is because the area of contact between the palm and the needle is very small therefore the pressure is large.

Whereas the area of contact between the pencil and the palm is more therefore the pressure is less.

Conclusion: Two factors which determine the magnitude of the pressure are:-

• Force – greater the force greater is the pressure and vice-versa.
• Coverage area –greater the area less is the pressure and vice-versa.

Fluid pressure

Normal force exerted by fluid per unit area. This means force is acting perpendicular to the surface of contact.

If a body is submerged in the water, force is exerted by the water perpendicular to the surface of the body. Fluid force exerts itself perpendicularly to any surface in the fluid, no matter the orientation of that surface. Thus, fluid pressure has no intrinsic direction of its own and can be considered as a scalar quantity.

Pressure is a scalar quantity. Because the force here is not a vector quantity but it is the component of force normal to the area.

Dimensional formula for pressure is[ ML^-1T²]. The S.I unit is Pascal (Pa).

Atmospheric pressure: The atmospheric pressure at a point is equal to the weight of the column of air of unit cross-sectional area extending from that point to the top of the atmosphere. Its value is 1.013 ×10⁵ Pa at sea level. Atmospheric pressure drops as altitude increases. It is measured using an instrument called a barometer.

Definition of 1 atm

An atmosphere (atm) is a unit of measurement equal to the average air pressure at sea level at a temperature of 15 degrees Celsius (59 degrees Fahrenheit). One atmosphere is 1,013 millibars, or 760 millimetres (29.92 inches) of mercury. Atmospheric pressure drops as altitude increases

Pascal’s law

Pascal’s law states that if the pressure is applied to uniform fluids that are confined, the fluids will then transmit the same pressure in all directions at the same rate.

Pascal’s law holds good only for uniform fluids.

Let us try to understand this with a suitable example. Consider a vessel of circular shape filled with water which has 4 openings and in the entire openings 4 pistons are attached.

• Apply force on the first piston; this piston will move inward and all other pistons will move outwards.

• This happens because when this piston moves inwards the pressure is exerted on the water. Water transmits this pressure in all the directions.
• The other pistons, except A, moves at the same speed which shows water has exerted pressure in all the directions

Conclusion:-

1. For a uniform fluid in equilibrium, pressure is the same at all points in a horizontal plane. This means there is no net force acting on the fluid; the pressure is the same at all the points.

1. A fluid moves due to the differences in pressure. That means fluid will always move from a point which is at a higher pressure to the point which is at a lower pressure.

Archimedes principle

Archimedes Principle:

• Consider a body partially or fully dipped in a fluid. The fluid exerts a contact force on this body. The resultant of all these contact forces is termed buoyant force or up thrust.
• F=weight of fluid displaced by the body
• This force is termed buoyant force and it acts vertically upwards (opposite to the weight of the body) through the centre of gravity of the displaced fluid. Mathematically,       

F=Vσg

Where V is the volume of displaced liquid and σ is the density of the liquid

• The apparent reduction in weight of body =Up thrust = weight of liquid displaced by the body.

Variation of pressure with depth

Consider a cylindrical object inside a fluid, consider two positions for this object. Fluid is at rest therefore the force along the horizontal direction is zero.

Force along the vertical direction

Consider two positions 1 and 2. Force at position 1 is perpendicular to cross-sectional area   A₁F₁ = P₁

Similarly,  A₂ F₂=P₂ 

Totalthe  force   F_net= F₁+F₂   as F₁ is along negative y axis and is negative and F₂ is along positive y axis and hence positive so we have now,

 F_net= -F₁+F₂ = -P₁A₁+P₂A₂=(P₂-P₁)A    , Taking A₁=A₂=A

This net force will be balanced by the weight of the cylinder. Therefore under equilibrium conditions.

Fnet= mg (weight of cylinder)=ρ Vg( weight of liquid displaced)

Fnet= (P₂-P₁) A=ρg(Ah)   using V= Ah

Which gives  (P₂-P_{1)A =ρghA   ⇒ (P2-P1)=ρgh}

Therefore the difference in the pressure is dependent on the height of the cylinder.

Consider the top of the cylinder exposed to air therefore P₁ = P_a (atmospheric pressure)

Then   P₂= P_a+ρgh

Conclusion:

•  The pressure P₂ , at depth  below the surface of a liquid open to the atmosphere is greater than atmospheric pressure by an amount ρgh. 
• The pressure is independent of the cross sectional or base area or the shape of the container.
• Thus, the pressure P, at depth below the surface of a liquid open to the atmosphere, is greater than atmospheric pressure by an amount ρgh. The excess of pressure, P − Pa, at depth h is called a gauge pressure at that point.

Hydrostatic Paradox

Hydrostatic Paradox means: - hydro = water, static =at rest

Paradox means that something is taking place surprisingly.

• Consider 3 vessels of very different shapes (like thin rectangular shape, triangular and some filter shape) and we have a source from which water enters into these 3 vessels.
• Water enters through the horizontal base which is the base of these 3 vessels. We observe that the level of water in all the 3 vessels is the same irrespective of their different shapes.
• This is because pressure at some point at the base of these 3 vessels is the same.
• The water will rise in all these 3 vessels till the pressure at the top is same as the pressure at the bottom.
• As pressure is dependent only on height therefore in all the 3 vessels the height reached by the water is the same irrespective of difference in their shapes.

This experiment is known as Hydrostatic Paradox.

Applications: Pascal’s law for transmission of fluid pressure

Hydraulic lift:-

Hydraulic lift is a lift which makes use of fluid.  For example: Hydraulic lifts that are used in car service stations to lift the cars.

Principle: -

• Inside a hydraulic lift there are 2 platforms, one has a smaller area and the other one has a larger area. It is a tube-like structure which is filled with uniform fluid.
• There are 2 pistons (P1 and P2) which are attached at both the ends of the tube. Cross-sectional area of piston P1 is A1 and piston P2 is A2.
• If we apply force F1 on P1, pressure gets exerted and according to Pascal’s law pressure gets transmitted in all the directions and same pressure gets exerted on the other end. As a result the Piston P2 moves upwards.

Advantage of using hydraulic lift is that by applying small force on the small area we are able to generate a larger force.

Mathematically ,  F₁= P₁A₁  and F₂= P₂A₂    

Since by Pascal’s law

Hydraulic Brakes

• Hydraulic brakes work on the principle of Pascal’s law.
• According to this law whenever pressure is applied on fluid it travels uniformly in all directions.
• Therefore when we apply force on a small piston, the pressure gets created which is transmitted through the fluid to a larger piston. As a result of this larger force, uniform braking is applied on all four wheels.
• As braking force is generated due to hydraulic pressure, they are known as hydraulic brakes.
• Liquids are used instead of gas as liquids are incompressible.

Effect of gravity on fluid pressure

We can start by stating the relationship between gravity and fluid pressure. We can define both of the terms. We can also write down the formula to find the fluid pressure and see if it is related to gravity. Gravity is a force existing between bodies.

The formula to find the fluid pressure is given by the formula,

P=ρgh

Where ρ is the density of the fluid, g is the acceleration due to gravity and h is the depth of the fluid level.

Fluid pressure is the pressure at a point within a fluid arising due to the weight of the fluid. Gravity is the universal force of attraction acting between all matters.

Therefore, according to the formula   P=ρgh  the pressure exerted by a fluid, is directly proportional to the specific gravity at any point and to the height of the fluid above the point.

A fun thing to do: Virtual lab

Below is the link of the simulation of under pressure

Under pressure, this simulation is to understand the concept of pressure.  It will help us understand how pressure varies with depth for different fluids.

What can we do in this simulation?

• We can choose the density of the liquid and can fill the liquid up to the height we want to fill it.
• We can also change the value of gravity and using the meter to measure pressure we can get the value of pressure in terms of Metric unit (Pa) and other units like ‘atm’. We just need to drag the meter and place it to the point where we want to measure the pressure.
• We can also on /off the atmospheric pressure in this simulation to see its effects.