NCERT Class 11 Physics Chapter 10 Notes Mechanical Properties of Fluids - Download PDF

Ever wondered how water flows in a river, how planes fly in the sky or how bubbles form? That’s all explained in Class 11 Physics Chapter 9 Mechanical Properties of Fluids. This chapter helps you to understand how liquids and gases behave when force is applied, which is very useful for exams like CBSE, JEE, and NEET. These NCERT notes are prepared by Careers360 experts faculty based on the latest CBSE syllabus.

In these Class 11 Physics Notes, you will learn important topics like how pressure varies with depth, why fluids flow the way they do (Bernoulli’s Principle), and what makes things float or sink. These NCERT Class 11 Physics Notes also include important formulas along with diagrams, simple explanations, and real-life examples to help you study easily and score better in the exam. Students can download these notes in PDF for free and use it for revision anytime.

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• NCERT Solutions for Class 11 Physics Chapter 9 Mechanical Properties of Fluids
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NCERT Class 11 Physics Chapter 9 Notes

What are Fluids?

• Fluids are substances that can flow when an external force is applied to them.

• Gases and Liquids are fluids.

• It does not have a finite shape rather, it takes the shape of the vessel that contains it.

• The normal force exerted by the liquid when it is at rest on a surface is called the thrust of liquid.

• SI unit of thrust = (N) Newton.

• The fluid mechanics is divided as follows

1. When the fluid is at rest - hydrostatics

2. When the fluid is in motion – hydrodynamics

Pressure

• The normal force applied by a liquid per unit area of the surface in contact is called the pressure of the liquid, also known as Hydrostatic pressure.

• Pressure applied by a liquid column, p = hρg

here, h = height of the liquid column,

ρ= density of liquid

g =acceleration because of gravity

Mean pressure on the walls of the container containing liquid up to height h = (hρg / 2).

Density

The density of any material is defined as its mass per unit its volume. If a fluid of mass m occupies a volume V, then its density is given as

Density =mV

Density is usually denoted by the symbol ρ. It is a positive scalar quantity. Its SI unit is $kg{m}^{-3}$ and its dimensions are [$M{L}^{-3}$].

Relative Density or Specific Gravity:

The relative density of a substance is defined as the ratio of the density of the substance to the density of water at 4∘C. The density of water at 4∘C is 1.0×${10}^3$ kg $m^{-3}$. Relative density is a unitless and dimensionless scalar quantity. It is always positive.

If the density of silver is $10.8\times {10}^3$ $kg{m}^{-3}$, then its relative density is 10.8 .
Density plays an important role in describing pressure exerted by fluids.

Pascal’s Law

• According to Pascal’s law at equilibrium, the pressure at any point enclosed in a container exerts the pressure and is transmitted equally in all directions and on the Walls of the container.

• The working principle of hydraulic lift, hydraulic press, and hydraulic brakes is based on Pascal’s law.

• Atmospheric Pressure

• The pressure that is exerted by the atmosphere on Earth is known as atmospheric pressure.

• It is almost 100000 N/m².

• At sea level, atmospheric pressure becomes equal to 76 cm of mercury column.

• Atmospheric pressure is given as "hdg" which is equal to that of 76 x 13.6 x 980 dyne cm^-2

• Atmospheric pressure can be measured in a torr or bar. 1 torr = 1 mm of mercury column. 1 bar = 10⁵ Pa

• The aneroid barometer is the instrument that is used to measure atmospheric pressure.

Variation of Pressure with Depth

As we go deeper into a fluid (like water), the pressure increases. This happens because the weight of the fluid above adds to the pressure below. The pressure at a certain depth in a liquid is given by the formula:

P=h⋅ρ⋅g

Where:

• P is the pressure,
• h is the depth,
• ρ (rho) is the density of the liquid,
• g is the acceleration due to gravity.

So, the deeper you go, the higher the pressure becomes. This is why deep-sea divers need special suits to withstand the increased pressure underwater.

Atmospheric Pressure and Gauge Pressure

Atmospheric Pressure is the pressure exerted by the weight of the air in the Earth's atmosphere. At sea level, it is approximately $1.013\times {10}^5$ Pa (Pascal) or 1 atmosphere (atm). This pressure is all around us, but we usually don't feel it because it is balanced by the pressure inside our bodies.

Gauge Pressure is the pressure measured relative to atmospheric pressure. It tells us how much more or less pressure is present in a system compared to the surrounding air pressure. It is given by:

Gauge Pressure = Absolute Pressure - Atmospheric Pressure

Hydraulic Machines

When an external pressure is applied to a confined liquid at rest, it is transferred undiminished to all parts of the liquid, and also to the walls of the vessel containing the liquid. This is Pascal's law for transmission of fluid pressure.
Let's understand this with the following illustration. Take a vessel having four openings A,B,C and D as shown. These openings are provided with frictionless, weightless and water tight pistons.

All the four openings are equally wide, having a cross section a (say). When the piston at A is pushed in with a force F, it exerts a pressure P=Fa on the enclosed fluid. It is found that we require the same force F on each piston B,C and D to keep them from moving. Thus, the pressure at each piston increases by Fa. It proves that external pressure applied to the enclosed fluid is transferred to all points undiminished. This law has many applications in daily life.

Streamline Flow

Till now we have studied about the fluids at rest.
The study of fluids in motion is called fluid dynamics.
When a fluid is in flow, its motion can either be smooth or irregular, depending on its velocity of flow.

When a liquid flows such that each particle of the liquid passing a given point moves along the same path and has the same velocity as its predecessor had at that point, the flow is called streamlined or steady flow.
The path followed by a fluid particle in steady flow is called a streamline.

Properties of Streamlines

(i) The tangent at any point of a streamline gives the direction of flow of fluid particle at that point.
(ii) In a steady flow, no two streamlines can cross each other, for if they do so, two tangents can be drawn at the point of intersection. It means the oncoming fluid particle can go either one way or the other. Thus, the flow would not be steady.

(iii) Fluid velocity remains constant at any point of a streamline, but it may be different at different points of the same streamline.
(iv) Fluid velocity is greater at the regions where streamlines are closely spaced. This can be proved from the equation of continuity that we discuss in the upcoming section.

Equation of Continuity

Consider again a fluid in steady flow. The map of its flow for a particular section can be shown by a bundle of streamlines as shown below. Here area of cross-section is greater at point Q than that at point P. Hence, streamlines are closely spaced at P than those at Q. Let the area of cross-section and fluid velocity at point P be AP and vP respectively. Let the corresponding quantities at point Q be AQ and vQ.
Therefore, the volume of fluid moving in at P, in a small time interval Δt

=APvPΔt

Similarly, the volume flowing out at Q , during the same interval Δt,

=AQvQΔt

By conservation of mass, for Δt
mass flowing in at P= mass flowing out at Q.

By conservation of mass, for Δt
mass flowing in at P= mass flowing out at Q.

⇒(APvPΔt)ρP=(AQvQΔt)ρQ[ρP= fluid density at P,ρQ= fluid density at Q]

For the flow of incompressible fluids

ρP=ρQ⇒APvP=AQvQ

This expression is known as equation of continuity. It is a statement of conservation of mass in flow of incompressible fluids.

Bernoulli's Principle

Bernoulli's Principle states that for a streamlined flow of an ideal (non-viscous and incompressible) fluid, the total mechanical energy of the fluid remains constant. This total energy includes the pressure energy, kinetic energy, and potential energy per unit volume.

$P+\frac{1}{2}\rho v^2+\rho gh=$ constant

Applications of Bernoulli's Principle:

• Lift in airplane wings
• Flow of blood in arteries
• Functioning of a carburetor or atomizer

Limitations of Bernoulli's Equation

1. The equation is valid only for incompressible fluids having streamline flow. It is because it does not take into account the elastic energy of the fluids.
2. It is assumed that no energy is dissipated due to frictional force exerted by different layers of fluid on each other.
3. It does not hold for non-steady flow. In such situation velocity and pressure constantly fluctuate with time.

Speed of Efflux: Torricelli's Law

Torricelli's Law gives the speed of a fluid flowing out of an orifice (small hole) in a tank under the force of gravity. It is based on Bernoulli's Principle and assumes an ideal fluid and streamline flow.

According to Torricelli's Law, the speed of efflux (i.e., speed at which the liquid leaves the hole) is given by:

v=2gh

Where:

• v= speed of efflux
• g= acceleration due to gravity
• h= height of the fluid column above the hole

Dynamic Lift

Dynamic lift is the upward force experienced by an object when it moves through a fluid (like air or water). This force is generated due to the difference in pressure on different parts of the object caused by its shape and motion.

Viscosity

When a fluid moves, it flows in the form of parallel layers. These layers exert a force on each other which tends to oppose their relative motion. This is similar to what a frictional force does when two solids in contact move or tend to move over each other.

The property of fluid which gives rise to such frictional force in them, is called viscosity.
The force arises when there is relative motion between the layers of a fluid.

• Viscos Force (F)= -η A(dv/dx)
  η =-F/A(dv/dx)
  here,
  (dv/dx) = rate of change of the velocity with distance and velocity gradient, A = area of cross-section and coefficient of viscosity.

• SI unit of η = Nsm^-2 or pascal-second or decapoise.

• Dimensional formula = [ML^-1T^-1].

• The coefficient of viscosity of different oils and its variation with temperature helps us in selecting an appropriate lubricant for a given machine. Viscosity occurs due to the transmission of momentum. The value of viscosity for an ideal liquid is zero. In some organic liquids, it is used in determining the molecular weight and shape of large organic molecules like proteins and cellulose.

Buoyancy

• When a body is partially or fully immersed in a fluid an upward force acts on it and that upward force is called buoyant force.

• This whole phenomenon is known as Buoyancy.

• The buoyant force acts at the centre of gravity of the liquid displaced by the immersed part of the body, that point is known as the center buoyancy.

Archimedes’ Principle

• According to Archimedes Principle, the liquid is displaced by the weight of the object when immersed fully or partially and that weight is equal to the loss in weight when immersed in the liquid.

• Let ‘T’ be the observed weight of a body with density σ, when it is fully immersed in the liquid of density p, then the original weight of that body, w = T / (1 – p / σ)

Laws of Flotation

• Flotation of the object on a liquid surface requires that the mass of the object is similar to the mass of the liquid which is displaced by the immersion part of the body.

• If we consider ‘W’ is the mass of a body and ‘w’ is the buoyant force acting on that body:

1. In the first case: W > w, the body will sink to the bottom of the liquid.

2. In the second case W < w, the body will float partially submerged in the liquid.

3. In the third case W = w, the body will float in liquid

Conditions applicable for the stability of the body

For stable equilibrium: The center of buoyancy of the body lies vertically above the center of gravity.

For unstable equilibrium: The center of buoyancy of the body lies vertically above the center of gravity.

For neutral equilibrium: The center of buoyancy of the body coincides with the center of gravity.

Stoke’s Law

Stoke’s Law says that when a small spherical body drops in a liquid column sometimes it reaches a constant velocity. This velocity is called terminal velocity. When a small spherical object drops in the liquid column with terminal velocity, then the viscous force acting on it is given by

F = 6πηrv

where, r = radius of the body, V = terminal velocity, and η = coefficient of viscosity.

This is called Stokes law; where,

∙ ρ = density of the body

∙ σ = density of the liquid,

∙ η = coefficient of viscosity of liquid and,

∙ g = acceleration due to gravity 1. If ρ > ρ0, the body falls downwards.

If ρ < p0, the body moves upwards with the constant velocity.

If p0 << ρ, then v = (2r² ρg/9η)

Importance of Stokes Law

1. It helps to find the electric charge by Millikan’s oil drop experiment.

2. It helps during parachuting..

3. This is the phenomenon behind the formation of clouds.

Surface Tension

• The property of the liquid by virtue of which the free surface of the liquid starts minimizing due to force of attraction in between the layers of the liquid at rest.

• The surface tension is the physical property of the liquid in which molecules of liquids draw each and every side.

• SI unit of surface tension = Nm^-1

• Dimensional formula = [MT^-2]

Surface Energy

The liquid does not have any shape and when placed in any container possesses some additional energy, and this energy is different from the molecules that are present in the bulk amount inside the liquid. This energy is present at the surface of the molecules.

The product of surface tension and change in surface area is termed as a change in surface energy, and all are under constant temperature conditions.

Surface Energy and Surface Tension

Surface tension is defined as the surface energy per unit area or the force per unit length acting in the plane of the interface between the plane of the liquid and any other substance.

Consider a thin soap film obtained in a rectangular frame as shown above. Let the end AB be free to slide along the horizontal part of the frame. As AB slides to right by a distance d, some extra work is to be done to increase the surface area and hence the surface energy of the film. Let F is the force required to push AB towards right. The work done on the film is:

W=Fd

This work done is stored in the film as the extra surface energy. Let the surface tension i.e., the surface energy per unit area for the given case is S. Then

S(2dl)=Fd⇒S=F2l

[Soap film has two free surfaces]

This expression gives the measure of surface tension. Sl unit of surface tension is $N{m}^{-1}$. Its dimensions are [ $M{T}^{-2}$ ]. The force acting per unit length in the plane of the interface between a liquid and any other surface is independent of the area of the interface.

Angle of Contact

The angle of contact is defined as the angle that the tangent to the liquid surface at the point of contact makes with the solid surface inside the liquid. The angle of contact depends on the nature of the solid and the liquid in contact. At the point of contact, the surface forces between the three media must be in equilibrium. See the figure given next.

Drops in Fluids:

• A drop forms when a small amount of liquid is held together by surface tension.
• The spherical shape of a drop minimizes surface area, making it the most energy-efficient form.
• The pressure inside a drop is slightly higher than the surrounding fluid due to its curved surface (as per Laplace's law).

Bubbles in Fluids:

A bubble is typically a gas enclosed in a thin film of liquid (like a soap bubble).
It has two surfaces (inner and outer), so the excess pressure inside is twice that of a single surface:

ΔP=4Tr

where T is surface tension and r is the radius of the bubble.

Capillary Rise

Capillary Rise is a phenomenon observed when a liquid climbs up a narrow tube against gravity. It happens due to the combination of cohesive forces (between liquid molecules) and adhesive forces (between liquid and tube walls).
When adhesive forces are stronger (like in water and glass), the liquid rises in the capillary tube.
The height of rise h is given by the formula:

h=2Tcos⁡θrρg

where:

• T= surface tension of the liquid
• θ= angle of contact
• r= radius of the tube
• ρ= density of the liquid
• g= acceleration due to gravity

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