Study Guide - Class XII Physics CHAPTER I: Force …physics-eclass.weebly.com/uploads/1/6/2/2/16227420/chapter_1... · Study Guide - Class XII Physics Prepared by Amit Dahal, YHSS

Study Guide - Class XII Physics

Prepared by Amit Dahal, YHSS Page 1 of 19

CHAPTER I: Force and Motion in Fluids [7%]

Fluid A fluid is substance (gas or liquid) that flows to assume the shape of the container in which it is placed and

deforms continuously under the application of a shear stress.

A shear stress, denoted τ (Greek: tau), is the component of stress coplanar with a material cross section. Shear

stress arises from the force vector component parallel to the cross section - pairs of equal and opposing forces

acting on opposite sides of an object.

Forces parallel to the area resisting the force cause shearing stress. Shear stress is also known as tangential stress.

Flow The movement of liquids and gases is generally referred to as "flow," a concept that describes how fluids behave

and how they interact with their surrounding environment — for example, water moving through a channel or

pipe, or over a surface. Flow can be either steady or unsteady.

Fluid Flow is a part of fluid mechanics and deals with fluid dynamics. Fluids such as

gases and liquids in motion are called as fluid flow. Motion of a fluid is subjected to

unbalanced forces. This motion continues as long as unbalanced forces are applied.

For example, if you are pouring a water from a mug, the velocity of water is very high

over the lip, moderately high approaching the lip, and very low at the bottom of the

mug. The unbalanced force is gravity, and the flow continues as long as water is

available and the mug is tilted.

Types of Fluid Flow Fluid flow has all kinds of aspects — steady or unsteady, compressible or

incompressible, viscous or non-viscous, and rotational or irrotational, to name a few. Some of these

characteristics reflect properties of the liquid itself, and others focus on how the fluid is moving.

Steady or Unsteady Flow: Fluid flow can be steady or unsteady, depending on the fluid’s velocity:

Steady: In steady fluid flow, the velocity of the fluid is constant at any point.

Unsteady: When the flow is unsteady, the fluid’s velocity can differ between any two points.

Viscous or Nonviscous Flow: Liquid flow can be viscous or nonviscous.

Viscosity is a measure of the thickness of a fluid, and very gloppy fluids such as motor oil or shampoo are

called viscous fluids.

Fluid Flow Rate

The volume of fluid replaced in a given interval of time is called the fluid flow rate.

Mass flow rate = ρAV

Where,

ρ = density

V = Velocity

A = area

Flow rate = Area × Velocity



Viscosity In case of steady flow of a fluid, when a layer of fluid slips or tends to slip on adjacent layers in contact; the two

layers exert tangential force on each other which tries to destroy the relative motion between them.

The property of a fluid due to which it opposes the relative motion between its different layers is called viscosity

(fluid friction or internal friction) and the force between the layers opposing the relative motion is the viscous

force or viscous drag.

Viscous drag acts tangentially on the layers of the fluid in motion. This force becomes apparent when a layer of

fluid is made to move in relation to another layer.

Greater the friction, greater the amount of force required to cause this movement, which is called shear.

Shearing occurs whenever the fluid is physically moved or distributed, as in pouring, spreading, spraying,

mixing, etc. Highly viscous fluids, therefore, require more force to move than less viscous materials.

Cause of viscosity

Viscosity is due to the internal frictional force that develops between different layers of fluids as they are forced

to move relative to each other. Viscosity is caused by the cohesive forces between the molecules in liquids, and

by the molecular collisions in gases. Liquids have higher dynamic viscosities than gases.

Coefficient of viscosity According to Newton’s hypothesis, viscous force is directly

proportional to the area of the plane and velocity gradient

𝑭 ∝ 𝑨 𝒂𝒏𝒅 𝑭 ∝𝒅𝒗

𝒅𝒙

∴ 𝑭 ∝𝒅𝒗

𝒅𝒙⇒ 𝑭 = −𝜼𝑨

𝒅𝒗

𝒅𝒙 (𝜼: 𝒄𝒐𝒆𝒇𝒇𝒊𝒄𝒊𝒆𝒏𝒕 𝒐𝒇 𝒗𝒊𝒔𝒄𝒐𝒔𝒊𝒕𝒚)

Negative sign is because viscous force acts in a direction opposite to

the fluid flow.

The difference in velocity between adjacent layers of the fluid is known as a velocity gradient and is given

by v/x, where v is the velocity difference and x is the distance between the layers. To keep one layer of fluid

moving at a greater velocity than the adjacent layer, a force F is necessary, resulting in a shearing

stress F/A, where A is the area of the surface in contact with the layer being moved.

If A = 1 and dv/dx = 1, then 𝜼 = 𝑭

Therefore, the coefficient of viscosity is defined as the viscous force acting per unit area between two layers

moving with unit velocity gradient.

Units: C.G.S System: dyne-s/cm2 or Poise; S.I System : Newton-s/m2 or Poiseuille

1 Poiseuille = 10 Poise

Dimension: [ML-1 T-1]



With increase in pressure, the viscosity of liquids increases (except water) while that of gases is independent of

pressure. In-case of water, viscosity decreases with increase in pressure.

Motion of bodies falling in a uniform Gravitational Field with fluid resistance

Objects falling through a fluid eventually reach terminal velocity, when the resultant force acting on them is zero

and they move at a steady speed.

Falling objects

Usually two forces affect the falling object

1. Weight of the object: Force acting downwards, caused by the object’s mass the Earth's gravitational field.

2. Air resistance: Frictional force acting in the opposite direction to the movement of the object.

When an object is dropped, its motion can be described in three stages

1. At the start, the object accelerates downwards because of its weight (w = mg). There is no air resistance.

There is a resultant force acting downwards.

𝐹𝑟𝑖𝑐𝑡𝑖𝑜𝑛𝑎𝑙 𝑓𝑜𝑟𝑐𝑒 𝑓 = −𝑘𝑣 = 0

2. As it gains speed, the object's weight stays the same, but the air resistance on it increases. There is a

resultant force acting downwards.

3. Eventually, the object's weight is balanced by the air resistance. There is no resultant force and the object

reaches a steady speed, called the terminal velocity (Vt).

Weight (w) = Frictional force or drag force (f)

𝑚𝑔 = −𝑘𝑣𝑡

∴ 𝑣𝑡 =𝑚𝑔

𝑘

So when the body attains terminal velocity, it no longer accelerates, so a = 0



Derivation of velocity under free fall with fluid resistance

Force with which body falls down is

𝑤 ∝ 𝑚𝑔 … … … (𝑖) [m: mass of the body and g: acceleration due to gravity]

For small speed and neglecting buoyancy viscous drag “f” is

𝑓 ∝ 𝑣

𝑓 = −𝑘𝑣 … … … (𝑖𝑖)

[v: velocity of falling object and k is the constant of proportionality]

Net force

𝐹𝑛𝑒𝑡 = 𝑤 + 𝑓 ⇒ 𝑚𝑎 = 𝑚𝑔 − 𝑘𝑣

𝑚𝑔 − 𝑘𝑣 = 𝑚𝑑𝑣

𝑑𝑡

𝑑𝑣

𝑚𝑔 − 𝑘𝑣=

𝑑𝑡

𝑚

∫𝑑𝑣

𝑚𝑔 − 𝑘𝑣= ∫

𝑑𝑡

𝑚

𝑡

0

𝑣

0

⇒ ∫1

𝑚𝑔 − 𝑘𝑣𝑑𝑣 =

1

𝑚∫ 𝑑𝑡

𝑡

0

𝑣

0

[ln (𝑚𝑔 − 𝑘𝑣)

−𝑘]

0

𝑣

=𝑡

𝑚… … … … … . (𝑠𝑖𝑛𝑐𝑒 ∫

1

𝑚𝑔 − 𝑘𝑣𝑑𝑣 = [

ln (𝑚𝑔 − 𝑘𝑣)

𝑑𝑑𝑣

(𝑚𝑔 − 𝑘𝑣)]

𝑣

0

)

ln (𝑚𝑔 − 𝑘𝑣)

−𝑘−

ln 𝑚𝑔

−𝑘=

𝑡

𝑚

𝑙𝑛 ((𝑚𝑔 − 𝑘𝑣)

𝑚𝑔) =

−𝑘𝑡

𝑚 ⇒

(𝑚𝑔 − 𝑘𝑣)

𝑚𝑔= 𝑒

−𝑘𝑡𝑚

1 −𝑘𝑣

𝑚𝑔= 𝑒

−𝑘𝑡𝑚 ⇒ 1 − 𝑒

−𝑘𝑡𝑚 =

𝑘𝑣

𝑚𝑔⇒

𝑚𝑔

𝑘(1 − 𝑒

−𝑘𝑡𝑚 ) = 𝑣

𝒗 = 𝒗𝒕 (𝟏 − 𝒆−𝒌𝒕𝒎 ) … . . 𝑤ℎ𝑒𝑟𝑒 𝒗𝒕𝑖𝑠 𝑡ℎ𝑒 𝑡𝑒𝑟𝑚𝑖𝑛𝑎𝑙 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦

Terminal Velocity (Vt)

It is defined as the maximum constant velocity acquired by a body while falling through a viscous medium.

Terminal Velocity depends on

i. Surface area of the falling object. The greater it is, the shorter the time it reaches terminal velocity

and hence smaller terminal velocity. ii. Mass of the falling object. The bigger it is, the longer the time it reaches terminal velocity and hence

larger terminal velocity.



Graphical representation of variation of velocity with time under free fall

Acceleration of body falling in a viscous fluid

We know that acceleration is

𝑎 =𝑑𝑣

𝑑𝑡… (𝑖) … . . (𝑟𝑎𝑡𝑒 𝑜𝑓 𝑐ℎ𝑎𝑛𝑔𝑒 𝑜𝑓 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦)

Also for a falling object under gravity with viscous drag, we know that

𝑣 = 𝑣𝑡 (1 − 𝑒−𝑘𝑡𝑚 ) … . (𝒊𝒊) … . . 𝑤ℎ𝑒𝑟𝑒 𝒗𝒕𝑖𝑠 𝑡ℎ𝑒 𝑡𝑒𝑟𝑚𝑖𝑛𝑎𝑙 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦

Differentiating eqn. (ii)

𝑑𝑣

𝑑𝑡=

𝑑

𝑑𝑡 𝑣𝑡 (1 − 𝑒

−𝑘𝑡

𝑚 )

𝑎 = 𝑣𝑡

𝑑

𝑑𝑡(1 − 𝑒

−𝑘𝑡𝑚 )

𝑎 = 𝑣𝑡 [𝑑

𝑑𝑡(1) −

𝑑

𝑑𝑡(𝑒

−𝑘𝑡𝑚 )]

𝑎 = 𝑣𝑡 [0 − (𝑒−𝑘𝑡

𝑚 .𝑑

𝑑𝑡(

−𝑘𝑡

𝑚))] … … … … . [

𝒅

𝒅𝒙𝒆𝒂𝒙 = 𝒆𝒂𝒙.

𝒅

𝒅𝒙𝒂𝒙]

𝑎 = 𝑣𝑡 [−𝑒−𝑘𝑡𝑚 .

−𝑘

𝑚

𝑑

𝑑𝑡𝑡]

𝑎 = 𝑣𝑡 [𝑘𝑒

−𝑘𝑡𝑚

𝑚. 1]

𝑎 = 𝑣𝑡 .𝑘𝑒

−𝑘𝑡𝑚

𝑚… … … … (𝒊𝒊𝒊)

We know that terminal velocity 𝑣𝑡 is given by

𝑣𝑡 =𝑚𝑔

𝑘

So replacing it in eqn. (iii)

𝑎 =𝑚𝑔

𝑘.𝑘𝑒

−𝑘𝑡𝑚

𝑚

𝒂 =𝑔. 𝒆−𝒌𝒕𝒎

Acceleration Vs time graph



Surface Tension The molecules on the surface of a liquid, that is, the interface between the liquid and the air are bound together

by a weak -force called surface tension. This force makes the liquid form a layer and is caused

due to the cohesive force between the molecules of the liquid.

Surface tension is the property of a liquid by virtue of which its free surface behaves like a

stretched membrane in order to acquire minimum surface area.

Imagine a line AB in the free surface of a liquid at rest as shown in the figure alongside.

The force of surface tension is measured as the force acting per unit length on either side of this imaginary line

AB.

The force is perpendicular to the line and tangential to the liquid surface.

If F is the force acting on the length L of the line AB, then surface tension is given by:

𝑻 =𝑭

𝑳

Properties of Surface Tension Scalar quantity

Temperature sensitive

Impurity sensitive

Depends on the nature of the liquid

Independent of the area of liquid surface

Surface tension at a molecular level

Water molecules cling to each other. At the surface, however, there

are fewer water molecules to cling to since there is air above (no water

molecules). This results in a stronger bond between those molecules

that actually do come in contact with one another, and a layer of

strongly bonded water. This surface layer (held together by surface

tension) creates a considerable barrier between the atmosphere and the

water.

Within a body of a liquid, a molecule will not experience a net force

because the forces by the neighboring molecules all cancel out.

However, for a molecule on the surface of the liquid, there will be a

net inward force since there will be no attractive force acting from

above. This inward net force causes the molecules on the surface to

contract and to resist being stretched or broken. Thus the surface is under tension, which is called "surface

tension.

Due to the surface tension, small objects will "float" on the surface of a fluid, as long as the object cannot break

through and separate the top layer of water molecules. When an object is on the surface of the fluid, the surface

under tension will behave like an elastic membrane.

Therefore, surface tension is defined as the force per unit length acting perpendicular on an imaginary line

drawn on the liquid surface, tending to pull the surface apart along the line.

S.I unit : N m-1

C.G.S unit : dyne cm-1

Dimensional formula : MT-2.



Some examples of surface tension

i. Small insects such as the water strider can walk on water because their weight is not enough to penetrate

the surface.

ii. A carefully placed small needle can be made to float on the surface of water even though it is several

times as dense as water. If the surface is agitated to break up the surface tension, then needle will quickly

sink.

iii. Common tent materials are somewhat rainproof in that the surface tension of water will bridge the pores

in the finely woven material. But if you touch the tent material with your finger, you break the surface

tension and the rain will drip through.

iv. Normal urine has a surface tension of about 66 dynes/centimeter but if bile is present (a test for jaundice),

it drops to about 55. In the Hay test, powdered sulfur is sprinkled on the urine surface. It will float on

normal urine, but will sink if the surface tension is lowered by the bile.

v. Disinfectants are usually solutions of low surface tension. This allows them to spread out on the cell

walls of bacteria and disrupt them.

vi. Soaps and detergents help the cleaning of clothes by lowering the surface tension of the water so that it

more readily soaks into pores and soiled areas.

vii. The reason for using hot water for washing is that its surface tension is lower and it is a better wetting

agent. But if the detergent lowers the surface tension, heating may be unnecessary.

viii. The surface tension of water provides the necessary wall tension for the formation of bubbles with water.

The tendency to minimize that wall tension pulls the bubbles into spherical shapes.

ix. Surface tension is responsible for the shape of liquid droplets. Although easily deformed, droplets of

water tend to be pulled into a spherical shape by the cohesive forces of the surface layer.



Surface Energy The molecules on the liquid surface experience net downward force. So to bring a molecule from the interior of

the liquid to the free surface, some work is required to be done against the intermolecular force of attraction,

which will be stored as potential energy of the molecule on the surface. The potential energy of surface molecules

per unit area of the surface is called surface energy.

Unit : S.I. Unit: Joule/m2 and C.G.S. Unit: erg/cm2

Dimension : [MT–2]

If a rectangular wire frame ABCD, equipped with a sliding wire LM

dipped in soap solution, a film is formed over the frame. Due to the

surface tension, the film will have a tendency to shrink and thereby,

the sliding wire LM will be pulled in inward direction. However, the

sliding wire can be held in this position under a force F, which is equal

and opposite to the force acting on the sliding wire LM all along its

length due to surface tension in the soap film.

If T is the force due to surface tension per unit length, then

𝐹 = 𝑇 × 2𝑙 Here, l is length of the sliding wire LM. The length of the sliding wire has been taken as 2l for the reason

that the film has got two free surfaces.

Suppose that the sliding wire LM is moved through a small distance x, so as to take the position L’ M’. In

this process, area of the film increases by 2𝑙 × 𝑥 (on the two sides) and to do so, the work done is given by

𝑊 = 𝐹 × 𝑥 = (𝑇 × 2𝑙) × 𝑥 = 𝑇 × (2𝑙𝑥) = 𝑇 × ∆𝐴

∴ 𝑊 = 𝑇 × ∆𝐴 [∆A = Total increase in area of the film from both the sides]

If temperature of the film remains constant in this process, this work done is stored in the film as its surface

energy.

From the above expression

𝑇 =𝑊

∆𝐴 𝑜𝑟 𝑇 = 𝑊 [𝑖𝑓 ∆𝐴 = 1]

Therefore, Surface tension may be defined as the amount of work done in increasing the area of the liquid

surface by unity against the force of surface tension at constant temperature.

Surface tension is numerically equal to work done in stretching a unit area.



Angle of Contact

Angle of contact between a liquid and a solid is defined as the angle enclosed between the tangents to the liquid

surface and the solid surface inside the liquid, both the tangents being drawn at the point of contact of the liquid

with the solid.

Or It is defined as the angle between the tangent to the liquid surface at the point of contact and the solid

surface inside the liquid.

Angle of contact

depends on nature of solid and liquid in contact.

decreases on increasing the temperature.

increases with soluble impurities.

decreases with partially soluble impurities.

Wetting is a phenomenon whereby the liquid deposited on a solid (or liquid) substrate spreads out.

If the angle of contact is acute, liquid wets the surface and if the angle of contact is obtuse, the liquid dose not

wet the surface.



Capillarity

The molecules of a liquid have two types of forces acting on them.

Cohesive force: The force among the molecules of the liquid only, and

Adhesive force: The force acting between the molecules of the liquid and some other substance.

When the adhesion between the liquid and the container wall is more than the cohesion among the liquid

molecules, the liquid sticks to the walls of the container and results in capillary rise. The opposite of this behavior

happens when the cohesion is more than the adhesion - the capillary level dips.

If a narrow tube (called capillary) is dipped in a liquid, it is found that the liquid in the capillary either ascends

or descends relative to the surrounding liquid. This phenomenon is called capillarity.

The cause of capillarity is the difference in pressures on two sides of (concave and convex) curved surface of

liquid.

Examples of capillarity:

i. Ink rises in the fine pores of blotting paper leaving the paper dry.

ii. A towel soaks water.

iii. Oil rises in the long narrow spaces between the threads of a wick.

iv. Wood swells in rainy season due to rise of moisture from air in the pores.

v. Ploughing of fields is essential for preserving moisture in the soil.

vi. Sand is drier soil than clay. This is because holes between the sand particles are not so fine as

compared to that of clay, to draw up water by capillary action.

Meniscus A meniscus is a curve on the surface of a liquid when it touches another material. It can be convex with is the

curving outward or bulge outward or Concave just the opposite.

Adhesion is responsible for a meniscus and this has to do in part with water's

fairly high surface tension. Water molecules are attracted to the molecules in

the wall of the glass beaker. And since water molecules like to stick together,

when the molecules touching the glass cling to it, other water molecules cling

to the molecules touching the glass, forming the concave meniscus. They'll

travel up the glass as far as water's cohesive forces will allow them, until

gravity prevents them from going further. In case of Mercury, because cohesive force is stronger than its

attraction to the glass (adhesive force), convex meniscus is formed.

< 90o

concave

meniscus.

Liquid wets the solid surface

= 90o

Plane meniscus.

Liquid does not wet the solid

surface.

> 90o

convex

meniscus.

Liquid does not wet the solid

surface.



Excess pressure inside a liquid drop

Let R be the radius and , the surface tension of a liquid drop. Assume that the radius of the drop is increased

by dR doing a work.

Let the radius of the drop increase from R to (R + dR)

Initial surface area = 4πR2

Final surface area

= 4π(R + dR)2 = 4π(R2 + 2RdR + dR2)

= 4πR2 + 8πRdR (since dR2 is very small, it is neglected)

Increase in surface area

= 4πR2 + 8πRdR - 4πR2 = 8πRdR

Work done in enlarging the drop

= Increase in surface energy

= Increase in surface area x Surface tension

= 8πRdRT ….. (where T is the surface tension)

Also work done

= Force x distance

= pressure x Area x Distance

= p x 4πR2 x dR

So p x 4πR2 x dR = 8πRdRT ∴ 𝒑 =𝟐𝑻

𝑹

Rise of liquid in a capillary tube: Ascent formula

Let

Surface tension : T

Angle of contact : θ

Radius of capillary : r

Circumference of the tube : 2πr

Volume of the cylinder : πr2h

Total force acting in the upward direction = Tcosθ x 2πr

With this force, the liquid keeps rising to the height h, till the

upward force is equal to the downward force (weight of the

liquid column)

Downward force : Weight of the liquid column of height h

= mg

= vρg …….(v: volume, ρ : density)

= A h ρg = πr2hρg ….( A: area, h: height)

When the liquid stops rising, then

Tcosθ x 2πr = πr2hρg

2Tcosθ = hrρg

𝒉 =𝟐𝑻𝒄𝒐𝒔𝜽

𝒓𝝆𝒈… … 𝑨𝒔𝒄𝒆𝒏𝒕 𝒇𝒐𝒓𝒎𝒖𝒍𝒂



Fluid Flow

Ideal fluid

An ideal fluid is a fluid that has several properties including the fact that it is:

1. Incompressible : Density remains unchanged.

2. Irrotational : Flow of liquid is smooth and without any turbulence.

3. Non-viscous (In-viscid) : Fluid doesn’t have internal friction (ƞ = 0)

4. Steady Flow : Velocity of fluid at any fixed point remains same with time.

Fluid Flow is a term used to denote the flow of a fluid through a pipe (or any other structure). It is applied in

many disciplines of engineering and technology.

Fluid dynamics is a branch of fluid mechanics that deals with the study of fluid flow.

Types of Fluid Flow

i. Streamline flow

Streamline is defined as the path, straight or curved, the tangent to which at any

point gives the direction of the flow of liquid at that point.

In streamline flow, each element of the liquid passing through a point travels

along the same path and with the same velocity as the preceding element passes

through that point. Two streamlines cannot cross each other.

Greater the crowding of streamlines at a

place, greater is the velocity of liquid

particles at that place.

ii. Laminar flow

If a liquid is flowing over a horizontal surface with a steady flow and moves in the form of layers of different

velocities which do not mix with each other, then the flow of liquid is called laminar flow.

In laminar flow, the velocity of liquid flow is always less than the critical velocity of the liquid. The laminar

flow is synonymously used with streamlined flow.

The common example of laminar flow would be the smooth flow of a viscous liquid through a tube or pipe as

shown below.

In that case, the velocity of flow varies from zero at the walls to a maximum along the centerline of the vessel.

Figure showing the laminar flow of fluid between two rectangular plates each of area A



iii. Turbulent flow

When a liquid moves with a velocity greater than its critical velocity, the motion of the particles of liquid

becomes disordered or irregular. Such a flow is called a turbulent flow.

In turbulent flow, the path and the velocity of particles of the liquid change continuously and haphazardly with

time from point to point.

An example of turbulent flow is the flow of river water. River water does not follow any regular flow pattern.

It moves in a zigzag and haphazard manner.

Critical velocity (Vc) and Reynold’s Number (Re)

Critical velocity is the velocity of fluid flow up to which its flow is streamlined and above which its flow

becomes turbulent.

Critical velocity (Vc) is

Directly proportional to coefficient of viscosity 𝜂

𝑽𝒄 ∝ 𝜼

Inversely proportional to the density of fluid 𝜌

𝑽𝒄 ∝ 𝟏

𝝆

Inversely proportional to the radius r of the tube

𝑽𝒄 ∝ 𝟏

𝒓

∴ 𝑽𝒄 ∝ 𝜼

𝝆𝒓

𝑽𝒄 = 𝑹𝒆

𝜼

𝝆𝒓

𝒘𝒉𝒆𝒓𝒆 𝑹𝒆 𝒊𝒔 𝒕𝒉𝒆 𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕 𝒐𝒇 𝒑𝒓𝒐𝒑𝒐𝒓𝒕𝒊𝒐𝒏𝒂𝒍𝒊𝒕𝒚 𝒄𝒂𝒍𝒍𝒆𝒅 𝑹𝒆𝒚𝒏𝒐𝒍𝒅′𝒔 𝒏𝒖𝒎𝒃𝒆𝒓

Reynolds Number

Reynolds number (Re) is a dimensionless (pure) number used to determine the nature of flow (laminar or

turbulent flow) of liquid through a pipe.

It is defined as the ratio of the inertial force per unit area to the viscous force per unit area for a flowing fluid.

𝑹𝒆 =𝑰𝒏𝒆𝒓𝒕𝒊𝒂𝒍 𝒇𝒐𝒓𝒄𝒆 𝒑𝒆𝒓 𝒖𝒏𝒊𝒕 𝒂𝒓𝒆𝒂

𝑽𝒊𝒔𝒄𝒐𝒖𝒔 𝒇𝒐𝒓𝒄𝒆 𝒑𝒆𝒓 𝒖𝒏𝒊𝒕 𝒂𝒓𝒆𝒂

Consider a narrow tube having a cross-sectional area A. Suppose a fluid flows through it with the velocity 𝒗

for a time interval 𝚫𝒕

Inertial force per unit area is given by 𝐹

𝐴=

𝑅𝑎𝑡𝑒 𝑜𝑓 𝑐ℎ𝑎𝑛𝑔𝑒 𝑜𝑓 𝑚𝑜𝑚𝑒𝑛𝑡𝑢𝑚

𝐴

∆(𝑚 × 𝑣)

∆𝑡 × 𝐴=

𝐴𝑣 ∆𝑡 𝜌 × 𝑣

∆𝑡 × 𝐴= 𝝆𝒗𝟐



where

m: mass of the fluid

v : velocity

𝜌 : density of the fluid

Viscous force per unit area is given by

𝜂 × 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑔𝑟𝑎𝑑𝑖𝑒𝑛𝑡 = 𝜂𝑣

𝐿

∴ 𝑹𝒆 =𝑰𝒏𝒆𝒓𝒕𝒊𝒂𝒍 𝒇𝒐𝒓𝒄𝒆 𝒑𝒆𝒓 𝒖𝒏𝒊𝒕 𝒂𝒓𝒆𝒂

𝑽𝒊𝒔𝒄𝒐𝒖𝒔 𝒇𝒐𝒓𝒄𝒆 𝒑𝒆𝒓 𝒖𝒏𝒊𝒕 𝒂𝒓𝒆𝒂=

𝝆𝒗𝟐

𝜼𝒗𝑳

=𝝆𝒗𝑳

𝜼

where

m: mass of the fluid

v : velocity

𝜌 : density of the fluid

If the value of Reynold’s number lies

Between 0 and 2000, the flow of liquid is streamlined or laminar.

Between 2000 and 3000, the flow of liquid is unstable changing from streamlined to turbulent flow.

Above 3000, the flow of liquid is turbulent.

Flow Rate and the Equation of continuity The flow rate of a fluid is the volume of fluid which passes through a surface in a given unit of time . It is

usually represented by the symbol Q.

Volumetric flow rate is defined as

Q = v x a,

where Q is the flow rate, v is the velocity of the fluid, and a is the area of the cross section of the space the

fluid is moving through. Volumetric flow rate can also be found with

Q=V/t

where Q is the flow rate, V is the Volume of fluid, and t is elapsed time.

Continuity

The equation of continuity works under the assumption that the flow-in will equal the flow-out.

Flow in = Flow out

Mass m = volume x density

m = Area of cross section x length x density

Mass of the fluid that flows through section 1in time ∆t is

m1 = A1 v1∆t ρ1

Mass of the fluid that flows through section 2in time ∆t is

m2 = A2 v2∆t ρ2

by conservation of mass

m1 = m2

A1 v1∆t ρ1 = A2 v2∆t ρ2



For incompressible fluid, ρ1 = ρ2

∴ A1 v1 = A2 v2 = constant

Equation of continuity states that during the streamlined flow of the non-viscous and incompressible fluid

through a pipe of varying cross-section, the product of area of cross section and normal fluid velocity (av)

remains constant.

Av = constant

Equation of continuity is based on the principle of conservation of mass.

We can observe the continuity equation's effect in a garden hose. The water flows through the hose and when it

reaches the narrower nozzle, the velocity of the water increases. Speed increases when cross-sectional area

decreases, and speed decreases when cross-sectional area increases. This is a consequence of the continuity

equation. If the flow Q is held constant, when the area “A” decreases, the velocity “V” must increase

proportionally. For example, if the nozzle of the hose is half the area of the hose, the velocity must double to

maintain the continuous flow.



Bernoulli's theorem It states that the sum of pressure energy, kinetic energy and potential energy per unit volume of an

incompressible, non viscous fluid in a streamlined flow remains constant.

P + ½ ρv2 + ρgh = constant

Proof

Consider a fluid of negligible viscosity moving with laminar flow, as shown below

Let the velocity, pressure and area of the fluid column be v1, P1 and a1 at X and v2, P2 and a2 at Y.

Let the volume bounded by X and Y move to X’ and Y’ where XX’ = L1, and YY’ = L2. If the fluid is

incompressible

A1v1 = A2v2

The work done by the pressure difference per unit volume = gain in K.E. per unit volume + gain in P.E per unit

volume.

Work done = Force x Distance = p x volume x distance…… [Pressure (F/A) x Volume (AxL) = F x L ]

Net work done per unit volume = P1 - P2

K.E. per unit volume = ½ mv2 = ½ Vρ v2 = ½ρv2

Therefore:

K.E. gained per unit volume = ½ ρ(v22 - v1

2)

P.E. gained per unit volume = ρg(h2 – h1)

where h1 and h2 are the heights of X and Y above the reference level.

Therefore:

P1 - P2 = ½ ρ(v12 – v2

2) + ρg(h2 - h1)

P1 + ½ ρv12 + ρgh1 = P2 + ½ ρv2

2 + rgh2

Therefore:

P + ½ ρv2 + ρgh = constant

For a horizontal tube h1 = h2 and so we have:

P + ½ ρv2 = a constant

This is Bernoulli's theorem. We see that if there is an increase in velocity there must be a decrease of pressure

and vice versa.

The Bernoulli’s principle is based on the principle of conservation of energy.



Poiseuille's Formula

Let the motion of liquid through a narrow horizontal tube be streamlined. The layer of liquid in contact with

the surface of a horizontal tube is at rest velocity of layers increases as we move towards the axis of tube as

shown.

Volume of liquid flowing out per second (V) through narrow horizontal tube is

i. directly proportional to the pressure difference between two ends of a tube.

V∝ p

ii. directly proportional to the fourth power of the radius of a tube.

V ∝ r4

iii. inversely proportional to the length of tube.

𝑉 ∝1

𝑙

iv. inversely proportional to the coefficient of

𝑉 ∝1

𝜂

Combining all

𝑉 ∝𝑝𝑟4

𝜂𝑙⇒ 𝑽 =

𝝅

𝟖

𝒑𝒓𝟒

𝜼𝒍

The value of proportionality is found to be π/8 from experiment. This is known as Poiseuille’s formula

Stroke’s Law

It states that the backward viscous force acting on a small spherical body of radius r moving with a uniform

velocity v through a fluid of viscosity ƞ is given by

F = 6πƞrv

When a spherical body is dropped into it, the viscous force comes into play which opposes the motion of a body.

As the velocity of falling body increases, viscous force acting on it also increases in that proportion. At one

instant of time the body starts moving with uniform velocity called terminal velocity in the medium.

Determination of Stroke Law by Dimensional Method Motion of a small spherical body through a viscous medium and viscous force (F) acting on the body is found

to depend upon following factors:

1. The coefficient of viscosity of the medium.

F ∝ η

2. The radius of a spherical body.

F ∝ r

3. The velocity of the body in the medium.

F ∝ v

Combining all

F ∝ ηrv

F = kηrv



Let F = kηa rb vc Writing dimensional formula on both sides we have,

[MLT−2] = [ML−1T−1]a [L]b [LT−1]c

[MLT−2] = [Ma L(−a+b+c) T−a−c]

Equating powers of M,L and T on both sides, we have

a =1

-a + b + c =1

-a – c = -2

On solving, a = b = c = 1

∴ 𝑭 =𝒌𝜼𝒓𝒗 ∴ 𝑭 =𝟔𝝅𝜼𝒓𝒗

Terminal velocity based on Stroke’s law

When any object falls through a viscous fluid the following forces acts on the body

i. Downward force (weight of the body)

𝐹𝑑 = 𝑚𝑔 = 𝑣𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑠𝑝ℎ𝑒𝑟𝑒 × 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 × 𝑎𝑐𝑐. 𝑑𝑢𝑒 𝑡𝑜 𝑔𝑟𝑎𝑣𝑖𝑡𝑦 =𝟒

𝟑𝝅𝒓𝟑𝝆𝒈

ii. Upward force (buoyant force) equal to the weight of the liquid displaced

𝐹𝑢 = 𝑣𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑓𝑙𝑢𝑖𝑑 × 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 × 𝑎𝑐𝑐. 𝑑𝑢𝑒 𝑡𝑜 𝑔𝑟𝑎𝑣𝑖𝑡𝑦 =𝟒

𝟑𝝅𝒓𝟑𝝈𝒈

iii. Viscous force acting in upward direction

𝑭 = 𝟔𝝅𝜼𝒓𝒗

When the body attains terminal velocity (vc),

𝑭𝒅 = 𝑭𝒖 + 𝑭 4

3𝜋𝑟3𝜌𝑔 =

4

3𝜋𝑟3𝜎𝑔 + 6𝜋𝜂𝑟𝑣𝑐

6𝜋𝜂𝑟𝑣𝑐 =4

3𝜋𝑟3𝜌𝑔 −

4

3𝜋𝑟3𝜎𝑔

6𝜋𝜂𝑟𝑣𝑐 =4

3𝜋𝑟3(𝜌 − 𝜎)𝑔

𝒗𝒄 =𝟐

𝟗

𝒓𝟐(𝝆 − 𝝈)𝒈

𝜼



Poisson’s ratio

Strain is defined as the ratio of change in dimension to original dimension of a body when it is deformed. It is a

dimensionless quantity as it is a ratio between two quantities of same dimension.

When a deforming force is applied at the free end of a suspended wire its length increases while its diameter

decreases. There are two forces involved in this case – Lateral and Longitudinal strain.

The strain perpendicular to the applied force is called lateral strain. - The ratio of change in

diameter/breadth to original diameter/breadth is called the lateral strain.

The strain in the direction of the applied force is the longitudinal strain - The ratio of change in length

to original length is called the longitudinal strain.

The ratio of the lateral strain to the longitudinal strain in a stretched wire is called Poisson’s ratio.

If the original diameter of the wire is d and the contraction of the diameter under

stress is Δd, the lateral strain is expressed as

∆𝑑

𝑑

If the original length of the wire is L and the elongation under stress is ΔL, the

longitudinal strain is expressed as ∆𝐿

𝐿

Poisson’s ratio is then expressed as

𝜂 =

∆𝑑𝑑

∆𝐿𝐿

= −𝚫𝒅𝑳

𝚫𝑳𝒅

Poisson’s ratio is a ratio of two strains; it is a pure number and has no dimensions

or units. Its value depends only on the nature of material.

Poisson's effect has a considerable influence is in pressurized pipe flow. When

the air or liquid inside a pipe is highly pressurized it exerts a uniform force on the inside of the pipe, resulting in

a stress within the pipe material. Due to Poisson's effect, this stress will cause the pipe to increase in diameter

and slightly decrease in length. The decrease in length, in particular, can have a noticeable effect upon the pipe

joints, as the effect will accumulate for each section of pipe joined in series. A restrained joint may be pulled

apart or otherwise prone to failure.

Documents

Study Guide - Class XII Physics CHAPTER I: Force …physics-eclass.weebly.com/uploads/1/6/2/2/16227420/chapter_1... · Study Guide - Class XII Physics Prepared by Amit Dahal, YHSS