Mechanical Properties of Fluids Class 11 Notes: CBSE Physics Chapter 9

1. Fluid Mechanics:

• The term ‘fluids’ are generally used to express both liquids and gases. In other words, it can be said that the substances which have the potential to flow are termed as fluids.

• Fluids are assumed to be 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).

1.1. Fluid Statics:
1.1.1. Fluid Pressure:

• Fluid pressure $$\text{p}$$ at each point is defined as the normal force acting per unit area. Mathematically, 

$$p=\frac{d{F}_l}{dA}$$

• The S.I. unit of pressure is the $$\text{pascal (Pa)}$$ and $$1\text{ p}ascal=1N/m^2$$

• 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:

1. Pressures at two positions in a horizontal plane or at an equal level when the fluid is at rest or moving with constant velocity are the same.

2. Pressures at two positions that are separated at a depth of $$\text{h}$$ when fluid is at rest of moving with constant velocity are related by the expression:$$p_2-p_1=\rho gh$$, where $$\rho$$ is the density of liquid.

3. Pressures at two positions in a horizontal plane when a fluid container is having a constant horizontal acceleration, are related by the expression: 

          $$p_1-p_2=l\rho a$$

Also, $$\tan \theta =a/g$$, where $$\theta$$ is the angle between the liquid’s free surface and the horizontal.

4. Pressures at two positions within a liquid at a vertical distance of $$\text{h}$$ when the liquid container is accelerating up are related by the expression: $$p_2-p{}_1=\rho (g+a)h$$

         When the container accelerates down, then 

$$p_2-p{}_1=\rho (g-a)h$$.

1.1.2. Atmospheric Pressure  

• The pressure of the atmosphere of the earth is termed as atmospheric pressure. Normal atmospheric pressure at sea level (an average value) is $$\text{1}$$ atmosphere (atm), which is equal to $$\text{1}\text{.013 }\times \text{ 1}{\text{0}}^{\text{5}}\text{Pa}$$.

• The pressure which is excess above the atmospheric pressure is termed gauge pressure, and the total pressure is termed absolute pressure.

• The barometer is a device utilised to measure the atmospheric pressure whereas a U–tube manometer or simply a manometer is a device utilised to measure the gauge pressure.

1.1.3. Pascal’s Law:  

• A pressure change, on its application to an enclosed fluid, is transmitted undiminished to every portion of the fluid and to the walls of the containing vessel.

• Among the many practical applications of Pascal’s law, a popular application is the hydraulic lift.

1.1.4. 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 upthrust.

• $$F=weightoffluiddisplacedbythebody$$ 

• This force is termed buoyant force and it acts vertically upwards (opposite to the weight of the body) through the center of gravity of the displaced fluid. Mathematically,        

$$F=V\sigma g$$

where V is the volume of displaced liquid and $$\sigma$$ is the density of the liquid 

• The apparent reduction in weight of body$$=$$upthrust$$=$$weight of liquid displaced by the body.

• Flotation:

1. A body is said to float in a liquid when the average density of the body is less than that of the liquid.

2. The weight of the liquid displaced by the immersed part of the body should be the same as the weight of the body.

3. The center of gravity of the body and center of buoyancy should be along the same vertical line.

1.2. Fluid Dynamics: 

• Steady Flow (Streamline Flow)

The kind of flow in which the velocity of fluid particles crossing a particular position is the same at all times. Clearly, each particle follows the same path as followed by the previous particle through that point.

• Line of flow

Line of flow can be referred to as the path followed by a particle in a flowing liquid. With respect to steady flow, it is termed streamline. Two streamlines are never said to intersect with each other.

1.2.1. Equation of Continuity: 

• In a time $$\text{t}$$, the volume of liquid entering the tube of flow in a steady flow is given by: $${\text{A}}_{\text{1}}{\text{V}}_{\text{1}}\mathrm{\Delta }\text{ t}$$. 

• The same volume should flow out of this tube since the liquid is incompressible in nature. The volume flowing out in $$\text{t}$$ is given by$${\text{A}}_{\text{2}}{\text{V}}_{\text{2}}\mathrm{\Delta }\text{ t}$$.

• In the given diagram, $$A_1{V}_1=A_2{V}_2$$ 

• Also, mass flow rate is given by $$\rho \text{ AV}$$, where $$\rho$$ is the density of the liquid.

1.2.2. Bernoulli’s Theorem:

• When a streamlined flow of an ideal fluid is considered, the sum of pressure energy per unit volume, potential energy per unit volume and kinetic energy per unit volume are always constant at all cross-sections of the liquid. Mathematically,

          $$p+\rho gh+\frac{\rho V^2}{2}=const.$$

• Bernoulli’s equation is applicable only for an incompressible steady flow of a fluid with no viscosity.

• Applications of Bernoulli’s Theorem.

1. Velocity of efflux

Considering the diagram above, let us determine the velocity with which liquid comes out of a hole at a depth h below the liquid surface.

Using Bernoulli’s theorem,

$$P_A+\frac{1}{2}\rho {V_A}^2+\rho g{h}_A=P_B+\frac{1}{2}\rho {V_B}^2+\rho g{h}_B$$

$$\Rightarrow P_{atm}+\frac{1}{2}\rho {V_A}^2+\rho gh=P_{atm}+\frac{1}{2}\rho V^2+0$$

(Notice that $$P_B=P_{atm}$$, because we have opened the liquid to the atmosphere)

$$\Rightarrow V^2={V_A}^2+2gh$$

Using the equation of continuity,

$$A{V}_A=aV$$

Where A is the area of the cross-section of the vessel and a is the area of the hole.

$$\Rightarrow V^2=\frac{a^2}{A^2}{V}^2+2gh$$

$$\Rightarrow V=\frac{\sqrt{2gh}}{\sqrt{1-a^2/A^2}}\approx \sqrt{2gh}$$ (when the hole is very small)

2. Venturi Meter

A venturi meter is an instrument used for measuring the rate of flow of fluids.

Considering the diagram above,

If $${\text{P}}_{\text{A}}$$ is pressure at $$\text{A}$$ and $${\text{P}}_{\text{B}}$$ is pressure at $$\text{B}$$,

 $$P_A-P_B=h\rho g$$ 

Where h is the difference of heights of liquids of density $$\rho$$ in vertical tubes. 

Now, when $${\text{V}}_{\text{1}}$$ is velocity at $$\text{A}$$ and $${\text{V}}_{\text{2}}$$ is velocity at $$\text{B}$$, using the equation of continuity,

$$Q=A_1{V}_1=A_2{V}_2$$;

Using Bernoulli’s theorem,

$$P_A+\rho \frac{{V_1}^2}{2}=P_B+\rho \frac{{V_2}^2}{2}$$  

$$\Rightarrow {V_2}^2-{V_1}^2=\frac{2}{\rho }(P_A-P{}_B)=\frac{2}{\rho }h\rho g$$

$$\Rightarrow \frac{Q^2}{{A_2}^2}-\frac{Q^2}{{A_1}^2}=2hg(Q=AV)$$

$$\Rightarrow Q=A_1{A}_2\sqrt{\frac{2hg}{{A_1}^2-{A_2}^2}}$$, which is an expression for the rate of flow of fluids using a venturi meter.

1.3. Viscosity:

• The characteristic of a fluid by virtue of which it opposes the relative motion between its different layers is termed viscosity and the force which comes into action is termed the viscous force. Mathematically, viscous force is given by:

$$F=-\eta A\frac{dv}{dx}$$

Where $$\eta$$ is a constant dependent on the nature of the liquid and is termed the coefficient of viscosity and $$\frac{dv}{dx}$$ is the velocity gradient.

     S.I. unit of coefficient of viscosity is $$Pa.sorNs{m}^{-2}$$.

      CGS unit of viscosity is $$poise(1Pa.s=10poise)$$.

1.3.1. Stoke’s Law:

• When a solid travels through a viscous medium, its motion is opposed by a viscous force dependent on the velocity, shape and size of the body.

• The viscous drag acting on a spherical body of radius $$r$$, travelling with velocity $$v$$, in a viscous medium of viscosity $$\eta$$ is given by $$F_{viscous}=6\pi \eta rv$$

This formula is termed Stoke’s law.

• Importance of Stoke’s law:

• This law is utilized in finding the electronic charge with the help of Millikan’s experiment.

• This law talks about the formation of clouds.

• This law talks about why the speed of raindrops is less than that of an object falling freely with a constant velocity from the height of clouds.

• This law allows a person to fly down with the help of a parachute.

1.3.2. Terminal Velocity:

• It refers to the maximum constant velocity acquired by the body under free fall in a viscous medium. Mathematically,  

$$v_r=\frac{2r^2(\rho -{\rho }_0)g}{9\eta }$$

1.3.3. Poiseuille’s Formula:

• Poiseuille learnt the stream line flow of liquid in capillary tubes.

The volume of liquid coming out of the tube per second is equal to $$\frac{\pi {Pr}^4}{8\eta l}$$.

1.3.4. Reynold Number:

• The maintenance of stability of laminar flow is facilitated by viscous forces. However, it is seen that laminar or steady flow is disrupted when the rate of flow is large. Irregular, unsteady motion and turbulence occur at high flow rates.

• Reynold’s number refers to a dimensionless number whose value provides an approximate idea of the flow rate, if it would be turbulent or not. Mathematically, it is given by

$$R_e=\frac{\rho vD}{\eta }$$

Where, 

$$\rho =thedensityofthefluidflowingwithaspeedv$$

$$D=thediameterofthetube$$

$$\eta =thecoefficientofvis\cos ityofthefluid$$

• It is observed that flow is streamline or laminar for $${\text{R}}_{\text{e}}$$ less than $$\text{1000}$$ whereas the flow is turbulent for $${\text{R}}_{\text{e}}>2000$$. Additionally, the flow becomes unsteady for $${\text{R}}_{\text{e}}$$ between $$\text{1000}$$ and $$\text{2000}$$.

1.4. Surface Tension:

• The surface tension of a liquid refers to the force per unit length in the plane of a liquid surface perpendicular to either side of an imaginary line drawn on that surface. Mathematically,

$$S=\frac{F}{l}$$ 

Where,

$$S=surfacetensionofliquid.$$

• Unit of surface tension in MKS system: $$N/morJ/m^2$$

Unit of surface tension in CGS system: $$dyne/cmorerg/c{m}^2$$

1.4.1. Surface Energy:

• In order to increase the surface area, work needs to be done over the surface of the liquid. This work is stored in the liquid surface as its potential energy. Thus, the surface energy of a liquid refers to the excess potential energy per unit area of the liquid surface. Mathematically,

$$W=S\mathrm{\Delta }A;where\mathrm{\Delta }A=increaseinsurfacearea.$$

1.4.2. Excess Pressure:

• Excess pressure in a liquid drop or bubble in a liquid is given by $$P=\frac{2T}{R}$$.

• Excess pressure in a soap bubble is given by $$P=\frac{4T}{R}$$, as it has two free surfaces.

1.4.3 Angle of Contact:

• The angle between the tangent to the liquid surface at the point of contact and the solid surface inside the liquid is termed the angle of contact $(\theta )$, as shown in the diagram in the following section.

• When the glass plate is immersed in mercury, the surface becomes curved and the mercury gets depressed below. The angle of contact turns out to be obtuse for mercury.

• When the plate is dipped in water with its side vertical, the water gets drawn up along the plane and forms of a curved shape. The angle of contact turns out to be acute for water.

1.4.4 Capillary Tube and Capillary Action:         

• A very narrow glass tube with a fine borehole and open at both ends is called a capillary tube. When a capillary tube is dipped in a liquid, the liquid rises or falls in the tube and this reaction is known as capillarity.

• Mathematically, capillary rise or fall (h) is given by

$$h=\frac{2S\cos \theta }{r\rho g}=\frac{2S}{R\rho g}$$

Where, 

$$S=surfacetension$$

$$\theta =angleofcontact$$

$$r=radiusofcapillarytube$$

$$R=radiusofmeniscus$$

$$\rho =densityofliquid$$

• The capillary rise in a tube of insufficient length:

When the actual height to which a liquid would rise in a capillary tube is ‘h’, then a capillary tube of length less than ‘h’ can be considered as a tube of “insufficient length”.

In this case, the liquid rises to the top of the capillary tube of length $$l(l<h)$$ and adjusts the radius of curvature of its meniscus till the excess pressure is equalised by the pressure of the liquid column of length $$\text{l}$$. (Notice that liquid does not overflow here). 

• Consider the following figure:

• Mathematically, 

$$\frac{2\sigma }{r^{\prime }}=l\rho g...(i)$$

where $r^{\prime }$ is the new curvature.

Now, when $$\text{r}$$ is the actual radius of curvature,

$$\frac{2\sigma }{r}=h\rho g...(ii)$$

          Comparing $$\text{(i) and (ii)}$$;

          $$\Rightarrow \frac{2\sigma }{\rho g}=l{r}^{\prime }=hr$$

$$\Rightarrow r^{\prime }=\frac{hr}{l}$$, which is an expression for the new curvature in the case of capillary rise in a tube of insufficient length.

• Capillarity for different cases of adhesion and cohesion:

Mechanical Properties of Fluids Class 11 Notes

Fluids and Their Properties

Fluids are substances that have no definite shape and can flow easily from higher levels to lower levels. For example, liquids and gases are fluids.

They don’t have a definite shape, and they take the shape of a container.

Properties of Fluid

1. Hydrostatic - Fluids at rest

2. Hydro-dynamic - Fluids in motion

Thrust

Thrust is a force exerted by the fluid at the walls of the container. The S.I. unit of thrust is Newton, and the CGS unit is Dyne.

Pascal’s Law

Pascal’s law states that if gravity’s effect is ignored, then pressure applied to any point in an enclosed incompressible fluid is transmitted equally in all directions throughout the fluid.

We know that   P1 - P2 = hρg

At g = 0,  P1 - P2 = 0 

⇒ P1 = P2

This means at g = 0, the pressure at every two points inside the liquid is the same.

Archimedes’ Principle

When a body or object is completely or partially immersed in a liquid, at rest, the body loses some of its weight. This apparent loss in the weight of the body is equal to the weight of the liquid displaced by the immersed part of the body.

Specific Gravity of the body

The specific gravity of the body is equivalent to the relative density of the body.

It is given by:

$$\frac{\text{Weight of body in air}}{\text{Loss of weight of the body in the water at }{4}^{\circ }C}$$

The Density of the Mixture of a Substance

1. If two liquids of different masses m1 and m2, and densities ρ1 and ρ2 are mixed, then the density of a mixture of the substances is given by:

             $${\rho }_{Mixture}=\frac{{\rho }_1{\rho }_2(m_1+m_2)}{m_1{\rho }_2+m_2{\rho }_1}$$

2. If the mass is the same, but densities are different, i.e.,  ρ1 and ρ2. Then ρMixture  is:

                                 $$=\frac{{\rho }_1+{\rho }_2}{2}$$

3. If the number of substances of volume V1, V2 and densities ρ1 and ρ2 are mixed, then the mixture’s density is:

                 $${\rho }_{Mixture}=\frac{{\rho }_1{V}_1+{\rho }_2{V}_2}{V_1+V_2}$$

Law of Floatation

When a body of density ρ, and volume V is immersed completely in a liquid, two forces act on it. If W is the weight of the body and w is the buoyant force, then three cases are possible. They are:

Case 1: W > w. In this case, the body will completely sink in the liquid.

Case 2: W < w. In this case, the body will float partially and be immersed partially in water.

Case 3: W = w. In this case, the body will float in the liquid in a balanced position.

Viscosity

• Stoke’s Law

• Terminal Velocity

• Poiseuille’s Formula

• Reynold Number

Fluid mechanics is filled with various observations conducted by eminent scientists. This section in class 11th Physics Chapter 9 Notes also includes some of the essential laws and principles you must prepare to score considerably in exams. Explanations, equations, SI and CGI units, etc. all are laid out concisely such that you do not miss out on anything while revising.

Surface Tension

• Surface Energy

• Excess Pressure

• Angle of Contact

• Capillary Tube and Capillary Action

The final section in the notes of Chapter 9 Physics Class 11 discusses all essential details related to surface tension. Furthermore, you will also study how to increase surface area, and how to determine excess pressure in a liquid bubble and angle of contact. Also, an experiment using a capillary tube is provided to rule out capillarity action and radius of curvature.

If you have exams around the corner, we recommend you to download notes on Mechanical Properties of Fluids Class 11 offered by Vedantu, immediately. Drafted by experienced Physics lecturers, this revision guide ensures to provide you with outstanding grades.

Thermal Properties of Matter Class 11 Notes Physics - Basic Subjective Questions

Section – A (1 Mark Questions)

1. Three bodies are having temperature $T_A=-{42}^{\circ }F,T_B=-{10}^{\circ }C,T_C=200K$ . Which body among these is most warm?

Ans. Let us calculate all body temperatures on the same scale

Let choose Celsius

C=K-273.15

C=(5F-160)/9

$T_A=-{42}^{\circ }F=(5\times -42-160)/9=-{41}^{\circ }C$

$T_B=-{10}^{\circ }C$

$T_C=200K=200-273\cdot 15=-73\cdot 1^{\circ }C$

So the most warm body is TB.

2. If a thermometer reads the freezing point of water as 20°C and the boiling point as 150°C, how much does the thermometer read, when the actual temperature is 60°C?

Ans. 150C-20C=100x

1x=1.3C

thermometer reading = 1.3C + 20

$=1\cdot 3(60)+20={98}^{\circ }C$

3. Burns from steam are usually more serious than those from boiling water. Why?

Ans. Steam will produce more severe burns than boiling water because steam has more heat energy than water due to its latent heat of evaporation.

For water, the latent heat of fusion and vaporisation is, $L_f=3\cdot 33\times {10}^{5}Jk{g}^{-1}$ and $L_V=22\cdot 6\times {10}^{5}Jk{g}^{-1}$ respectively. It means $22\cdot 6\times {10}^{5}Jk{g}^{-1}$ of heat is needed to convert 1 kg of water to steam at 100°C. So, steam at 100°C carries more heat than water at 100°C. This is why burns from steam are usually more serious than those from boiling water.

4. A bimetallic strip consists of brass and iron when it is heated it bends into an arc with brass on the convex and iron on the concave side of the arc. Why does this happen?

Ans. Two strips of equal lengths but of different materials (different coefficient of linear expansion $\propto$ ) when joined together, are called a bimetallic strip. This strip has the characteristic property of bending on heating due to the unequal linear expansion of the two metals. The brass side bends on the outer side (convex side) due to greater $\propto$ and the iron bends on the inner side (concave side) due to smaller $\propto$.

5. Why are birds often seen to swell their feathers in winter?

Ans. When birds swell their feathers, they trap air in the feathers. Air being a poor conductor prevents loss of heat and keeps the bird warm.

Solved Sample Numerical Problem

Q1. A vessel 6m high is half-filled with water and then filled to the top with oil of density 0.96 g/c.c. What is the pressure at the bottom of the vessel due to these liquids?

Given:  h = 6 m,  ρ0 = 0.96 g/c.c. = 0.96 x 103kgm-3,ρW = 103kgm-3

To find:  P    

Solution: Let’s find out the mean density. It is given by:

                    $${\rho }_{Mean}=\frac{{\rho }_w+{\rho }_0}{2}=\frac{{10}^3+0.96\times {10}^3}{2}=0.98\times {10}^{3}kg{m}^{-3}$$

Therefore, the pressure at the bottom is given by:

                                 P = hρg

$$\Rightarrow P=6\times 0.98\times {10}^3\times 9.8=5.7624\times {10}^{4}N{m}^{-2}$$.

For downloading the notes on mechanical properties of fluids class 11, click on the link given on this page for Class 11 Mechanical Properties of Fluids Notes.

Important formula in Class 11 Physics Chapter 9 Mechanical Properties of Fluids

1. Pressure (P):
$P=\frac{F}{A}$
Where F is the force and A is the area over which the force is applied.

2. Pascal’s Law:
$\mathrm{\Delta }P=\rho gh$
Where $\mathrm{\Delta }P$ is the change in pressure, $\rho$ is the fluid density, g is the acceleration due to gravity, and h is the height of the fluid column.

3. Buoyant Force (Archimedes’ Principle):
$F_b=\rho gV$
Where $F_b$​ is the buoyant force, ρ\rhoρ is the density of the displaced fluid, g is the acceleration due to gravity, and V is the volume of the displaced fluid.

4. Equation of Continuity:
$A_1{v}_1=A_2{v}_2$
Where $A_1$​ and $A_2$ are the cross-sectional areas at points 1 and 2, and v1v_1v1​ and $v_2$ are the fluid velocities at these points. This expresses the conservation of mass in fluid flow.

5. Bernoulli’s Equation:
$P+\frac{1}{2}\rho v^2+\rho gh=\text{constant}$
This is a statement of energy conservation for fluid flow, relating pressure P, fluid velocity V, and height h at different points.

6. Viscosity (Stokes' Law):
$F=6\pi \eta rv$
Where F is the force due to viscosity, $\eta$ is the coefficient of viscosity, r is the radius of the sphere, and v is the velocity.

7. Surface Tension (TTT):
$T=\frac{F}{l}$
Where F is the force acting on the surface of the liquid, and l is the length over which the force acts.

8. Capillary Rise:
$h=\frac{2T\cos \theta }{\rho gr}$​
Where h is the height of the liquid column, T is the surface tension, $\theta$ is the contact angle, $\rho$ is the density of the liquid, and r is the radius of the tube.

Important Topics of Class 11 Physics Chapter 9 Mechanical Properties of Fluids

Importance of Class 11 Physics Chapter 9 Mechanical Properties of Fluids Revision Notes 

• They break down complex terms like viscosity and streamline flow, helping students develop a strong understanding of fluid mechanics, which is essential for both exams and practical applications.

• The notes emphasise important laws, such as Archimedes’ principle and Stokes' law, which are vital in solving numerical problems related to buoyancy and fluid resistance.

• By focusing on the derivation of key formulas, these notes make it easier to understand the underlying physics concepts rather than just memorising equations.

• They help clarify the distinction between different fluid behaviours, like laminar and turbulent flow, which is important for understanding fluid dynamics in various systems.

• The revision notes also include solved examples and common exam questions, aiding in practice and reinforcing the application of theoretical concepts.

• By concisely summarising core concepts, they serve as a quick reference guide, making them ideal for last-minute revision.

• They support the development of analytical skills, encouraging students to approach real-world problems related to fluid mechanics with a solid foundation of knowledge.

Tips for Learning the Class 11 Chapter 9 Physics Mechanical Properties of Fluids

• Start by understanding the basic concepts of pressure, buoyancy, and viscosity. These are fundamental to the chapter and form the foundation for more complex topics.

• Break down the formulas and understand their derivations rather than just memorising them. Knowing the logic behind Bernoulli’s principle, Pascal’s law and Archimedes’ principle will make it easier to apply them to problems.

• Visualise fluid flow using diagrams or animations. This will help in understanding concepts like streamlined flow, turbulent flow, and fluid dynamics.

• Practice solving numerical problems on buoyancy, surface tension, and fluid pressure. These are common areas in exams, and practising different scenarios will enhance your understanding.

• Relate the concepts to real-life examples, such as how aeroplanes fly (Bernoulli’s principle) or why objects float in water (Archimedes’ principle). This helps solidify the learning by making it relatable.

• Revise regularly and make use of short notes to remember key points and formulas, especially when preparing for exams.

• Work through past question papers and sample problems related to fluid mechanics to test your understanding of the subject.

Conclusion 

Class 11 Physics Chapter 9: Mechanical Properties of Fluids notes help simplify complex topics like pressure, viscosity, and buoyancy. By breaking down essential principles such as Bernoulli’s and Pascal’s laws, these notes make it easier to understand fluid behaviour in everyday situations. With well-organised content, key formulas, and examples, students can build a strong foundation in fluid mechanics. These notes also aid in exam preparation, providing a clear and concise way to review critical concepts and apply them in solving real-world problems related to fluid flow and pressure.

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