Link-1 Chapter 11 Thermal Properties of Matter Handwritten notes Class 11 Physics

Link-2 Chapter 11 Thermal Properties of Matter Handwritten notes Class 11 Physics

Thermal Properties of Matter Class 11 notes Physics Chapter 11

Introduction

In this chapter, we shall examine some of the thermal properties of matter. We will first consider thermal expansion which plays an important role in everyday life and then discuss changes of phase and latent heat. At the end, we will discuss the phenomenon of heat transfer.

When a body is heated, various changes take place. Temperature is a measure of ‘hotness’ of a body. A kettle with boiling water is hotter than a box containing ice. When water boils or freezes, its temperature does not change during these processes even though a great deal of heat is flowing into or out of it.

Temperature and Heat

(i). Temperature

Temperature is a relative measure, or indication of hotness or coldness. A hot cooker is said to have a high temperature, and ice cube to have a lower temperature. An object at a higher temperature is said to be hotter than the one at a lower temperature. The SI unit of temperature is kelvin (K), whereas degree celsius (°C) is a commonly used unit of temperature.

(ii). Heat

When you put a cold spoon into a cup of hot coffee, the spoon warms up and the coffee cools down as they were trying to equalise the temperature. Energy transfer that takes place solely because of a temperature difference is called heat flow or heat transfer and energy transferred in this way is called heat. The SI unit of heat energy transferred is expressed in joule (J).

Measurement of Temperature

A physical property that changes with temperature is called a thermometric property. When a thermometer is put in contact with a hot body, the mercury expands, increasing the length of the mercury column.

(i). Celsius Scale

It defines ice-point temperature as 0°C and the steam point temperature as 100°C. The space between 0°C and 100°C marks is equally divided into 100 intervals.

(ii). Fahrenheit Scale

It defines the ice-point temperature as 32°F and the steam point temperature as 212°F. The space between 32°F and 212°F is divided into 180 equal intervals.

(iii). Kelvin Scale

Kelvin Scale is a scale of measuring of temperature, the melting point of ice is taken as 273 K and the boiling point of water as 373 K the space between these two points is divided into 100 equal intervals.

(iv). Relation between Different Scales of Temperatures

To convert a temperature from one scale to the other, we must take into account the fact that zero temperatures of the two scales are not the same.

C100=F−32180=K−273100=R80C100=F-32180=K-273100=R80

Note: The normal temperature of the human body measured on the Celsius scale is 37°C which is 98.6°F.

Ideal Gas Equation and Absolute Temperature

(i). Ideal Gas Equation

An equation which follows the law of Boyal, law of Charls and llaw of Avogadro is called ideal gas equation.

At constant temperature,

V∝1PV∝1P …( From Boyal’s law)

At constant pressure,

V∝TV∝T …( From charl’s law)

At constant T and P,

V∝nV∝n …( From Avogadro’s law)

By combinig all above equation, we get

V∝TnPV∝TnP

V=nRTPV=nRTP

PV=nRTPV=nRT

where, n = Number of moles of gas

R = Universal gas constant (R = 8.31 J mol–1 K–1)

P = Pressure of gas

V = Volume of gas

(ii). Absolute Temperature

The absolute minimum temperature is equal to –273.15ºC. This is also known as absolute zero. Absolute zero is the foundation of the kelvin temperature scale or absolute scale temperature.

Thermal Expansion

Increase in size of any matter on heating is called thermal expansion. There are three types of thermal expansion.

(i). Linear Expansion

The expansion in length is called linear expansion and the fractional change in length, ΔL/L is given by ΔL/L = αΔT where α is called coefficient of linear expansion.

(ii). Area Expansion

The expansion in area is called area expansion or superficial expansion and the fractional change in area, ΔA/A is given by ΔA/A = βΔT where β is called coefficient of area expansion.

(iii). Volume Expansion

The expansion in volume is called volume expansion and the fractional change in area, ΔV/V is given by ΔV/V = γΔT where γ is called coefficient of volume expansion.

(iv). Relation Between

α : β : γ = 1 : 2 : 3

αβ=12αβ=12 …..[ β = 2α ]

αγ=13αγ=13 …..[ γ = 3α ]

Specific Heat Capacity

If an amount of heat Q, when given to a body of mass m, increases its temperature by an amount ΔT, then

Q = mcΔT

where c is a constant and is called the specific heat capacity or simply specific heat of the material of the body.

If m = 1 kg and ΔT = 1C° then c = Q

Specific heat of the material of a substance is the amount of heat required to raise the temperature of unit mass of the substance through 1C°.

In SI, the unit of c is J/kg K.

Calorimetry

Calorimetry deals with the measurement of heat. The vessel which is largely used in such a measurement is called a calorimeter.

When two bodies at different temperatures are allowed to share heat, they attain a common temperature. If it is assumed that no heat is received from or given to any body outside the system and if there is no chemical action involved in the process of sharing, then

Heat gained = Heat lost

This simple statement based on the law of conservation of energy is called the principle of calorimetry.

Change of State

Depending on temperature and pressure, all matter can exist in a solid, liquid or gaseous state. These states or forms of matter are also called the phases of matter.

The change of state from solid to liquid is called melting and from liquid to solid is called fusion. It is observed that the temperature remains constant until the entire amount of the solid substance melts. That is, both the solid and the liquid states of the substance coexist in thermal equilibrium during the change of states from solid to liquid.

The temperature at which the solid and the liquid states of the substance is in thermal equilibrium with each other is called its melting point. The change of state from liquid to vapour (or gas) is called vaporisation. It is observed that the temperature remains constant until the entire amount of the liquid is converted into vapour.

The temperature at which the liquid and the vapour states of the substance coexist is called its boiling point. The change from solid state to vapour state without passing through the liquid state is called sublimation, and the substance is said to sublime.

Latent Heat

Latent heat is defined as the heat or energy that is absorbed or released during a phase change of a substance. It could either be from a gas to a liquid or liquid to solid and vice versa. Latent heat is related to a heat property called enthalpy. It is denoted by L and its SI unit is J/kg.

L=QmL=Qm

There are two types of latent heat.

(i). Latent Heat of melting

It is a amount of heat which is required to change of phase from solid to liquid for unit mass at constant temperature. Ex- Latent heat of melting of ice is 3.33 x 10⁵ J/kg.

(ii). Latent Heat of Vaporization

It is a amount of heat which is required to change of phase from liquid to vapor for unit mass at constant temperature. Ex- Latent heat of vaporization of water is 22.6 x 10⁵ J/kg.

Heat Transfer

There are three mechanisms of heat transfer which name is given as- conduction, convection and radiation. Conduction occurs within a body or between two bodies in contact. Convection depends on motion of mass from one region of space to another. Radiation is heat transfer by electromagnetic radiation, such as sunshine, with no need for matter to be present in the space between bodies.

(i). Conduction

Conduction is the mechanism of transfer of heat between two adjacent parts of a body because of their temperature difference. Suppose, one end of a metallic rod is put in a flame, the other end of the rod will soon be so hot that you cannot hold it by your bare hands.

Here, heat transfer takes place by conduction from the hot end of the rod through its different parts to the other end. Gases are poor thermal conductors, while liquids have conductivities intermediate between solids and gases.

(ii). Convection

Convection is a mode of heat transfer by actual motion of matter. It is possible only in fluids. Convection can be natural or forced. In natural convection, gravity plays an important part. When a fluid is heated from below, the hot part expands and, therefore, becomes less dense. Because of buoyancy, it rises and the upper colder part replaces it. This again gets heated, rises up and is replaced by the relatively colder part of the fluid. The process goes on.

In forced convection, material is forced to move by a pump or by some other physical means. The common examples of forced convection systems are forced-air heating systems in home.

(iii). Radiation

Radiation is the transfer of heat by electromagnetic waves such as visible light, infrared, and ultraviolet rays. Everyone has felt the warmth of the sun’s radiation and intense heat from a charcoal grill or the glowing coals in a fireplace. Most of the heat from these bodies reaches you not by conduction or convection in the intervening air but by radiation. This heat transfer would occur even if there were nothing but vacuum between you and the source of heat.

Black Body Radiation

(i). Emissive Power

The amount of heat energy rediated per unit area of the surface of a body, per unit time and per unit wavelength range is constant which is called as the ’emissive power’ (e_λ) of the given surface, given temperature and wavelength. Its S.I. unit is Js^-1 m-^-2.

(ii). Absorptive Power

The ‘absorptive power’ of a surface at a given temperature and for a given wavelength is the ratio of the heat energy absorbed by a surface to the total energy incident on it at a certain time. It is represented by (a_λ). It has no unit as it is a ratio.

(iii) Perfect Black Body

A body is said to be a perfect black body, if its absorptivity is 1. It neither reflects nor transmits but absorbs all the thermal radiations incident on it irrespetive of their wavelengths.

(iv) Wein’s Displacement Law

This law states that as the temperature increases, the maximum value of the radiant energy emitted by the black body, move towards shorter wavelengths. Wein found that “The product of the peak wavelength (λ_m) and the Kelvin temperature (T) of the black body should remain constant.”

λm×T=bλm×T=b

Where b is constant known as Wein’s constant. Its value is 2.898 x 10^-3 mk.

(v) Stefan’s Law

This law states that the thermal radiations energy emitted per second from the surface of a black body is directly proportional to its surface area A and to the fourth power of its absolute temperature T.

Emission coefficient or degree of blackness of a body is represented by a dimensionless quantity ε, 0 < ε < 1. If ε = 1 then the body is perfectly black body. Hence

E∝AT4E∝AT4

E=σAT4E=σAT4

Where σ is a Stefan’s constant and its value is 5.67 x 10^-8 W m^-2 K^-4.

Newton’s Law of Cooling

According to Newton’s law of cooling, “The rate of loss of heat of a body is directly proportional to the excess of the temperature (T–T₀) of the body with respect to the surroundings”.

−dTdt∝(T−T0)-dTdt∝(T-T0)

Summary

• Temperature : The relative measure of hotness or coldness of a body is called its temperature.
• Heat : The energy that flows between two bodies by virtue of temperature difference between them is called heat. It flows from a hot body to cold body.
• Specific Heat Capacity : The amount of heat per unit mass absorbed or rejected by a substance to change its temperature by one unit is called its specific heat capacity (C).
• Molar Specific Heat Capacity : The amount of heat per unit mole absorbed or rejected by a substance to change its temperature by one unit is called its molar specific heat capacity (C).
• Calorimeter : A device in which heat measurement can be made is called a calorimeter.
• Melting Point : The temperature at which the solid and the liquid states of a substance exist in thermal equilibrium with each other is called its melting point.
• Boiling Point : The temperature at which the liquid and the vapour states of the substance coexist is called its boiling point.
• Triple Point : The temperature and pressure at which all the three phases of a substance co exist is called its triple point.
• Latent Heat : The heat per unit mass required (absorbed or evolved) to change the state of a substance at the same temperature and pressure is called its latent heat.
• The temperature of an object is measured with a device called thermometer.
• Heat transfer can take place by three modes namely, conduction, convection and radiation. Radiation is fastest of them all and does not require a material medium.

Chapter 10 Mechanical Properties of Fluids Handwritten notes Class 11 Physics

Mechanical Properties of Fluids Class 11 notes Physics Chapter 10

Introduction

In this chapter, we shall study some common physical properties of liquids and gases. How are fluids different from solids? What is common in liquids and gases? Unlike a solid, a fluid has no definite shape of its own. Solids and liquids have a fixed volume, whereas a gas fills the entire volume of its container.

Liquids and gases can flow and are therefore, called fluids. Earth has an envelop of air and two-thirds of its surface is covered with water. All the processes occurring in living beings including plants are mediated by fluids.

Pressure

We can define pressure as the normal force acting per unit area of a surface. It is denoted by P and SI unit of it is pascal (Pa). It is a scalar quantity.

P=FAP=FA

We observe that the same force (weight) exerts different pressures for different areas in contact. Lesser the area, more is the pressure exerted for a given force.

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=mVDensity=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 [ML^–3].

Pascal’s Law

This law states that the pressure in a fluid at rest is same at all points which are at the same height.

Imagine a small element of fluid in the shape of a right angled prism. All its points lie at the same depth inside the liquid. Therefore the effect of gravity is same at all these points. The forces exerted by rest of the fluid on different surfaces of this fluid element are as follows:

F1=F2=F3F1=F2=F3

If A₁, A₂ and A₃ are the surface areas of faces respectively, then by dividing.

F1A1=F2A2=F3A3F1A1=F2A2=F3A3

p1=p2=p3p1=p2=p3

So, the pressure exerted is same in all directions in a fluid at rest, at the points at equal height.

Variation of Pressure with Depth

Let’s find the difference in pressures at two points, whose levels differ by a height h in a fluid at rest. Let P₁ and P₂ be the pressures at two points 1 and 2 inside a fluid. Point 1 is at a height h above the point 2.

Imagine a fluid element in the shape of a cylinder as shown. If A be the area of the top and the bottom of this cylinder, then

F1=P1AF1=P1A

F2=P2AF2=P2A

Since, the fluid remains at rest, therefore the force F₂, which acts upwards should balance the two downward forces. These are, the force F₁ exerted at the top of the cylinder, and the weight W of the fluid confined within the cylinder.

F2=F1+mgF2=F1+mg

If ρ is the density of the fluid, then

F2=F1F2=F1+ρ.(volume of the cylinder).g

P2A=P1A+ρ.(A.h).gP2A=P1A+ρ.(A.h).g

P2=P1+ρghP2=P1+ρgh

P2−P1=ρghP2-P1=ρgh

This result tells us that as we go deep down a liquid the pressure goes on increasing. This pressure depends only on the height of the liquid column above the point.

Atmospheric Pressure

The atmospheric pressure at a point is equal to the weight of a 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 is measured using an instrument called barometer.

Units os Atmospheric Pressure

• SI unit of pressure is N m^–2 or Pascal (Pa)
• Atmosphere, 1 atm = 1.013 × 10⁵ Pa = 760 mm of Hg
• 1 torr = 133 Pa
• 1 mm of Hg = 1 torr
• 1 bar = 10⁵ Pa
• 1 millibar = 100 Pa

Gauge Pressure

When we remove atmospheric pressure from total pressure of any system then this remaining pressure is called Gauge Pressure. The excess pressure P–P_a, at depth h is called a gauge pressure at that point.

Archimedes’ Principle

When a body is partially or completely immersed in a liquid, it loses some of its weight. The loss in weight of the body in the liquid is equal to the weight of the liquid displaced by the immersed part of the body. The upward force excerted by the liquid displaced when a body is immersed is called buoyancy. Due to this, there is apparent loss in the weight experienced by the body.

Law of Floatation

A body floats in a liquid if weight of the liquid displaced by the immersed portion of the body is equal to the weight of the body. When a body is immersed partially or wholly in a liquid, then the various forces acting on the body are

1. Upward thrust (T) acting at the centre of buoyancy and whose magnitude is equal to the weight of the liquid displaced
2. The weight of the body (W) which acts vertically downward through its centre of gravity.(a) When W > T, the body will sink in the liquid;(b) When W = T, then the body will remain in equilibrium inside the liquid;(c) When W < T, then the body will come upto the surface of the liquid.

Streamline 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 streamline.

Equation of Continuity

According to this theorem, “For the streamline flow of an incompressible fluid through a pipe of varying cross-section, product of cross-section area and velocity of streamline flow (Av) remains constant throughout the flow”.

Av = constant

Bernoulli’s Principle

It may be stated as follows: As we move along a streamline, the sum of the pressure (P), the kinetic energy per unit volume (ρv22ρv22) and the potential energy per unit volume (ρgh) remains a constant.

P+ρgh+12ρv2=P+ρgh+12ρv2=constant

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.

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. The surface tension of a liquid usually decreases with increase in temperature.

S=FlS=Fl

Capillary Rise

A tube of very fine bore is called a capillary. ‘Capilla’ is a Latin word which means hair. Thus, capillary is a very thin tube. When such a tube, open at both ends, is dipped in a beaker containing water, water rises in it against gravity.

Let us find an expression for the height h, upto which a liquid rises in a capillary tube. Let a capillary tube of radius r be dipped in a liquid of surface tension S and density ρ.

Thus, capillary rise

h=2Scosθrρgh=2Scosθrρg

If the angle of contact for this liquid and the capillary tube is acute, the liquid forms a concave meniscus.

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. It is denoted by ‘η’. Its SI unit is N s m^–2 or Pa s which is also called poiseuille (Pl). The dimensions of viscosity are [ML^–1T^–1].

η=FAdvdxη=FAdvdx

where, dv/dx = velocity gardient

F = frictional force between layer of water

A = area of layer

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.

Stokes’ Law

When a spherical ball is dropped in a liquid, he observed that the viscous force F experienced by the ball is proportional to the

– velocity of the object through the fluid, v

– viscosity of the fluid, η

– radius of the sphere, r

Thus    F ∝ ηrv

Here, the constant of proportionality is found to be 6π.

Thus    F = –6πηrv   ← Stokes’ law.

The negative sign in the above expression just indicates that the retarding force is opposite to the direction of motion of the object.

Terminal Velocity

The maximum constant velocity acquired by a body while falling through a viscous medium is called its terminal velocity. It is usually denoted by V_T.

When the body acquires terminal velocity,

the upward viscous force + the upward buoyant force = weight of the ball

Reynolds Number

Whether a flow will be turbulent or not, is decided by a dimensionless parameter called Reynolds number R_e. This parameter is given by the relation

Re=ρvdηRe=ρvdη

where, ρ = density of the fluid

v = velocity of the fluid

η = viscosity of the fluid

d = diameter of the pipe through which the fluid flows

• If R_e < 1000, the flow is streamline or laminar
• If Re > 2000, the flow becomes turbulent
• If 1000 < Re < 2000, flow is unsteady i.e., it may change from laminar to turbulent and vice versa.

Poiseuille’s Formula

The Poiseuille’s formula gives an expression for volume flow rate through a capillary tube of inner radius r and length l due to a pressure difference between its ends, P. The volume flow rate is represented by Q such that

Q=dVdtQ=dVdt

According to Poiseuille’s,

dVdt=(π8)×(Pl)×r4ηdVdt=(π8)×(Pl)×r4η

Here, η is the coefficient of viscosity and all symbols have standard meaning.

Chapter 9 Mechanical Properties of Solids Hand Written Notes Class 11 Physics

Chapter 9 Mechanical Properties of Solids Class 11 notes Physics

Introduction

Can we design an aeroplane which is very light but sufficiently strong? Can we design an artificial limb which is lighter but stronger? Why does a railway track have a particular shape like I? Why is glass brittle while brass is not?

In this chapter, we will introduce the concepts of stress, strain and elastic modulus and a simple principle called Hooke’s law that help us predict what deformation will occur when forces are applied to a real kind of body.

Elastic Behaviour of Solids

In a solid, each atom or molecule is surrounded by neighbouring atoms or molecules. These are bonded together by interatomic or intramolecular forces and stay in a stable equilibrium position. When a solid is deformed, the atoms or molecules are displaced from their equilibrium positions causing a change in the interatomic distance.

When the deforming force is removed, the interatomic forces tend to drive them back to their original position. Thus the body regains its original shape and size.

(i) Deforming Force

If a force applied on a body produces a change in the normal positions of the molecules of the body, it is called deforming force.

(ii) Elasticity

The property of the body due to which, it tries to regain its original configuration when the deforming forces are removed is called elasticity.

(iii) Perfectly Elastic body

A body which completely regains its original configuration after the removal of deforming force, is called perfectly elastic body. Quartz and phosphor bronze are closest to perfectly elastic body known.

(iv) Perfectly Plastic Body

A body which does not regain its original configuration at all on the removal of deforming force, how so ever small the deforming force may be is called perfectly plastic body. For example, clay behaves like a perfectly plastic body.

(v) Restoring Force

When a deforming force is applied to a body to change its shape, the body develops an opposing force due to its elasticity. This opposing force tries to restore the original shape of the body, it is called restoring force.

Stress and Strain

(A) Strain

The strain is the relative change in dimensions of a body resulting from the external forces.

Strain = change in length / original length

It is a fractional quantity so, it has no unit.

(i) Tensile Strain

The tensile strain of the object is equal to the fractional change in length, which is the ratio of the elongation Δl to the original length l.

Tensile Strain=ΔlL=ΔlL

(ii) Shear Strain

We define shear strain as the ratio of the displacement x to the transverse dimension L.

Shear Strain=xL=xL

(iii) Bulk Strain

The fractional change in volume that is, the ratio of the volume change ΔV to the original volume V is called Bulk Strain.

Bulk Strain=ΔVV=ΔVV

(B) Stress

The restoring force developed per unit area in a body is called stress.

Stress = Restoring Force / area

In SI system, stress is measured in N / m² (pascal) and in CGS system in dyne/cm². The dimensional formula for stress is [M L^–1T^–2]

(i) Tensile Stress

We define the tensile stress at the cross-section as the ratio of the force F_⊥ to the cross-sectional area A.

Tensile Stress=F⊥A=F⊥A

(ii) Shear Stress

We define the shear stress as the force F_|| acting tangent to the surface, divided by the area A on which it acts.

Shear Stress=F∣∣A=F∣∣A

(iii) Bulk Stress

If an object is immersed in a fluid (liquid or gas) at rest, the fluid exerts a force on any part of the surface of the object. This force is perpendicular to the surface. The force F_⊥ per unit area that the fluid exerts on the surface of an immersed object is called the pressure p in the fluid (Bulk Stress).

Hooke’s Law

For small deformations the stress and strain are proportional to each other. This is known as Hooke’s law.

Thus,

stress ∝ strain

stress = k × strain

where k is the proportionality constant and is known as modulus of elasticity.

Elastic Moduli

The ratio of stress and strain, called modulus of elasticity, is found to be a characteristic of the material.

(i) Young’s Modulus

For a sufficiently small tensile stress, stress and strain are proportional. The corresponding elastic modulus is called Young’s modulus, denoted by Y.

Y = Tensile-stress / Tensile-strain=FAΔlL=FAΔlL

Y=FLAΔlY=FLAΔl

(ii) Shear Modulus

If the forces are small enough that Hooke’s law is obeyed, the shear strain is proportional to the shear stress. The corresponding elastic modulus is called the shear modulus, denoted by G. It is also called the modulus of rigidity.

G = Shear-stress / Shear-strain=FAΔxL=FAΔxL

G=F×LA×ΔxG=F×LA×Δx

G=FAθG=FAθ

SI unit of shear modulus is Nm^–2 or Pa.

(iii) Bulk Modulus

When Hooke’s law is obeyed, an increase in Bulk stress produces a proportional Bulk strain. The corresponding elastic modulus (ratio of stress to strain) is called the Bulk modulus, denoted by B.

When the pressure on a body changes by a small amount Δp, from p to (p+Δp), and the resulting Bulk strain is ΔV/V, Hooke’s law takes the form

B = Normal-stress / Volume-strain=−ΔpΔVV=-ΔpΔVV

We include a minus sign in this equation because an increase of pressure always causes a decrease in volume. The Bulk modulus B itself is a positive quantity.

SI unit of bulk modulus is Nm^–2 or Pa.

Compressibility

The reciprocal of the Bulk modulus is called the compressibility and is denoted by K. From equation

K=1BK=1B

The units of compressibility are those of reciprocal pressure, Pa^–1 or atm^–1.

Elastic Potential Energy

The excess of the energy of interaction between all atoms/molecules of a deformed object is elastic energy. When we remove the external force the body becomes undeformed and the elastic energy, will be retrieved back and converted into vibrational energy followed by heat, light, sound etc.

The elastic potential energy

ΔU=12=12stress × strain × volume

Also, the elastic potential energy per unit volume, i.e.,

ΔUvolume=12ΔUvolume=12stress × strain

Poisson’s Ratio

When a body is linearly extended, it contracts in the direction at right angles. Poisson’s ratio, σ is the ratio of the lateral strain to the longitudinal strain.

Longitudinal strain = Δl/L

Lateral strain = – ΔR/R

The Poisson’s ratio is given as,

σ = lateral strain / longitudinal strain

σ=–ΔRRΔlLσ=–ΔRRΔlL

–ve sign shows that if the length increases, then the radius of wire decreases. Poisson’s ratio is a unit less and dimensionless quantity.

Relation between Y, K, η and σ

• Y = 3K (1 – 2σ)
• Y = 2η (1 + σ)
• σ = (3K – 2η) /(2η + 6K)
• 9/Y = 1/K + 3/η

Applications of Elastic Behaviour of Materials

In our daily life, most of the materials which we use, undergo some kind of stress. That is why, while designing a structure of the material we give due consideration to the possible stresses, the material might suffer at one stage or the other. The following examples illustrate this concept.

1. The metallic parts of the machinery are never subjected to a stress beyond elastic limit, otherwise they will get permanently deformed.
2. The crane which is used to lift and move the heavy load is provided with thick and strong metallic ropes to which the load to be lifted is attached. The rope is pulled by using pulleys and motor.
3. The bridges are designed in such a way that they do not bend much or break under the load of heavy traffic, force of strongly blowing wind and its own weight.
4. Maximum height of a mountain on earth can be estimated from the elastic behaviour of earth. At the base of mountain, the pressure is given by p = ρgh, where h is the height of mountain, ρ is the density of material of mountain and g is the acceleration due to gravity.

Summary

• Elasticity : Elasticity is that property of the material of a body due to which the body opposes any change in its shape and size when deforming forces are applied on it and recovers its original configuration partially or wholly as soon as the deforming forces are removed.
• Stress : It is defined as the internal restoring force per unit area of cross-section of object.
• Strain : The change in dimensions of an object per unit original dimensions is called strain.
• Hooke’s law : For small deformation, the stress is proportional to strain.
• Young’s modulus : The ratio of tensile (or compressive) stress to the corresponding longitudinal strain is called Young’s modulus.
• Bulk modulus : The ratio of volumetric stress to volumetric strain is called Bulk modulus.
• Shear modulus : It is the ratio of shear stress to shearing strain.
• Poisson’s ratio : The lateral strain is proportional to longitudinal strain within the elastic limit and the ratio of two strains is called Poisson’s ratio.
• Elastic after effect : The slow process of recovering the original state by an object after the removal of the deforming force is called elastic after effect.

Chapter 8 Gravitation Handwritten Notes Class 11 notes Physics

Chapter 8 Gravitation Class 11 notes Physics

Summary

• Gravitational force: It is a force of attraction between the two bodies by the virtue of their masses.
• Acceleration due to gravity: The acceleration produced in the motion of a body freely falling towards earth under the force of gravity is known as acceleration due to gravity.
• Gravitational potential energy: The amount of work done in displacing the particle from infinity to a point under consideration.
• Gravitational potential: The gravitational potential due to the gravitational force of the earth is defined as the potential energy of a particle of unit mass at that point.
• Escape speed: The minimum speed with which the body has to be projected vertically upwards from the surface of the earth is called escape speed.
• Orbital speed: The minimum speed required to put the satellite into the given orbit around earth is called orbital speed.
• Satellite: It is a body which revolves continuously in an orbit around a comparatively much larger body.
• Polar satellite: It is the satellite which revolves in polar orbit around the earth.
• Geostationary satellite: It is the satellite which appears at a fixed position and at a definite height to an observer on earth.
• Kepler’s Ist law: All planets move in elliptical orbits, with the sun at one of foci of the ellipse.
• Kepler’s IInd law: The line that joins any planet to the sun sweeps out equal areas in equal intervals of time.
• Kepler’s IIIrd law: The square of the time period of revolution of a planet is proportional to the cube of the semi-major axis of the ellipse traced out by the planet.
• Newton’s universal law of gravitation: Every particle in the universe attracts every other particle with a force which is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.
• Gravitational force is a conservative force.
• The value of acceleration due ot gravity is maximum at the surface of the earth while zero at the centre of earth.
• Henry Cavendish was the first person who found the value of G experimentally.
• Gravitational force on a particle inside a spherical shell is zero.
• Gravitational shielding is not possible.
• An astronaut experiences weightlessness in a space satellite. It is because both the astronaut and the satellite are in “free fall” towards the earth.
• The value of g increases from equator to poles.
• The escape speed from a point on the surface of the earth may depend on its location on the earth e.g., escape speed is more on poles and less on equator.
• The orbital speed of satellite is independent of mass of the satellites.
• Kepler’s laws hold equally well for satellites.

Chapter 7 Systems of Particles and Rotational Motion Handwritten Notes Class 11 Notes Physics

Chapter 7 Systems of Particles and Rotational Motion Class 11 Notes Physics

Chapter 6 Work, Energy and Power Handwritten Notes Class 11th Physics

Chapter 6 Work, Energy and Power Class 11 Notes Physics

Chapter 5 Law of Motion Handwritten Notes Class 11th Physics

Chapter 5 Law of Motion Class 11 Notes Physics

 Dynamics is the branch of physics in which we study the motion of a body by taking into consideration the cause i.e., force which produces the motion.

Force

Force is an external cause in the form of push or pull, which produces or tries to produce motion in a body at rest, or stops/tries to stop a moving body or changes/tries to change the direction of motion of the body.
• The inherent property, with which a body resists any change in its state of motion is called inertia. Heavier the body, the inertia is more and lighter the body, lesser the inertia.
• Law of inertia states that a body has the inability to change its state of rest or uniform motion (i.e., a motion with constant velocity) or direction of motion by itself.

Newton’s Laws of Motion

Law 1. A body will remain at rest or continue to move with uniform velocity unless an external force is applied to it.
First law of motion is also referred to as the ‘Law of inertia’. It defines inertia, force and inertial frame of reference.
I here is always a need of ‘frame of reference’ to describe and understand the motion of particle, lhc simplest ‘frame of reference’ used are known as the inertial frames.
A frame of referent, e is known as an inertial frame it, within it, all accelerations of any particle are caused by the action of ‘real forces’ on that particle.
When we talk about accelerations produced by ‘fictitious’ or ‘pseudo’ forces, the frame of reference is a non-inertial one.

Law 2. When an external force is applied to a body of constant mass the force produces an acceleration, which is directly proportional to the force and inversely proportional to the mass of the body.

Law 3. “To every action there is equal and opposite reaction force”. When a body A exerts a force on another body B, B exerts an equal and opposite force on A.

Linear Momentum

The linear momentum of a body is defined as the product of the mass of the body and its velocity.

 Impulse

Forces acting for short duration are called impulsive forces. Impulse is defined as the product of force and the small time interval for which it acts. It is given by

Impulse of a force is a vector quantity and its SI unit is 1 Nm.
— If force of an impulse is changing with time, then the impulse is measured by finding the area bound by force-time graph for that force.
— Impulse of a force for a given time is equal to the total change in momentum of the body during the given time. Thus, we have

Law of Conservation of Momentum

The total momentum of an isolated system of particles is conserved.
In other words, when no external force is applied to the system, its total momentum remains constant.

Recoiling of a gun, flight of rockets and jet planes are some simple applications of the law of conservation of linear momentum.

Concurrent Forces and Equilibrium

“A group of forces which are acting at one point are called concurrent forces.”
Concurrent forces are said to be in equilibrium if there is no change in the position of rest or the state of uniform motion of the body on which these concurrent forces are acting.
For concurrent forces to be in equilibrium, their resultant force must be zero. In case of three concurrent forces acting in a plane, the body will be in equilibrium if these three forces may be completely represented by three sides of a triangle taken in order. If number of concurrent forces is more than three, then these forces must be represented by sides of a closed polygon in order for equilibrium.

Commonly Used Forces

(i) Weight of a body. It is the force with which earth attracts a body towards its centre. If M is mass of body and g is acceleration due to gravity, weight of the body is Mg in vertically downward direction.
(ii) Normal Force. If two bodies are in contact a contact force arises, if the surface is smooth the direction of force is normal to the plane of contact. We call this force as Normal force.

Example. Let us consider a book resting on the table. It is acted upon by its weight in vertically downward direction and is at rest. It means there is another force acting on the block in opposite direction, which balances its weight. This force is provided by the table and we call it as normal force.

(iii) Tension in string. Suppose a block is hanging from a string. Weight of the block is acting vertically downward but it is not moving, hence its weight is balanced by a force due to string. This force is called ‘Tension in string’. Tension is a force in a stretched string. Its direction is taken along the string and away from the body under consideration.

Simple Pulley

Consider two bodies of masses m₁ and m₂ tied at the ends of an in extensible string, which passes over a light and friction less pulley. Let m₁ > m₂. The heavier body will move downwards and the lighter will move upwards. Let a be the common acceleration of the system of two bodies, which is given by

Apparent Weight and Actual Weight

— ‘Apparent weight’ of a body is equal to its ‘actual weight’ if the body is either in a state of rest or in a state of uniform motion.
— Apparent weight of a body for vertically upward accelerated motion is given as
Apparent weight =Actual weight + Ma = M (g + a)
— Apparent weight of a body for vertically downward accelerated motion is given as
Apparent weight = Actual weight Ma = M (g – a).

• Friction

The opposition to any relative motion between two surfaces in contact is referred to as friction. It arises because of the ‘inter meshing’ of the surface irregularities of the two surfaces in contact.

Static and Dynamic (Kinetic) Friction

The frictional forces between two surfaces in contact (i) before and (ii) after a relative motion between them has started, are referred to as static and dynamic friction respectively. Static friction is always a little more than dynamic friction.
The magnitude of kinetic frictional force is also proportional to normal force.

Limiting Frictional Force
This frictional force acts when body is about to move. This is the maximum frictional force that can exist at the contact surface. We calculate its value using laws of friction.

Laws of Friction:

(i) The magnitude of limiting frictional force is proportional to the normal force at the contact surface.

(ii) The magnitude of limiting frictional force is independent of area of contact between the surfaces.

Coefficient of Friction

The coefficient of friction (μ) between two surfaces is the ratio of their limiting frictional force to the normal force between them, i.e.,

Angle of Friction

It is the angle which the resultant of the force of limiting friction F and the normal reaction R makes with the direction of the normal reaction. If θ is the angle of friction, we have

Angle of Repose

Angle of repose (α) is the angle of an inclined plane with the horizontal at which a body placed over it just begins to slide down without any acceleration. Angle of repose is given by α = tan-1 (μ)

Motion on a Rough Inclined Plane

Suppose a motion up the plane takes place under the action of pull P acting parallel to the plane.

•Centripetal Force
Centripetal force is the force required to move a body uniformly in a circle. This force acts along the radius and towards the centre of the circle. It is given by

where, v is the linear velocity, r is the radius of circular path and ω is the angular velocity of the body.

Centrifugal Force
Centrifugal force is a force that arises when a body is moving actually along a circular path, by virtue of tendency of the body to regain its natural straight line path.
The magnitude of centrifugal force is same as that of centripetal force.

Motion in a Vertical Circle

The motion of a particle in a horizontal circle is different from the motion in vertical circle. In horizontal circle, the motion is not effected by the acceleration due to gravity (g) whereas in the motion of vertical circle, the motion is not effected by the acceleration due to gravity (g) whereas in the motion of vertical circle, the value of ‘g’ plays an important role, the motion in this case does not remain uniform. When the particle move up from its lowest position P, its speed continuously decreases till it reaches the highest point of its circular path. This is due to the work done against the force of gravity. When the particle moves down the circle, its speed would keep on increasing.

Let us consider a particle moving in a circular vertical path of radius V and centre o tide with a string. L be the instantaneous position of the particle such that

Here the following forces act on the particle of mass ‘m’.
(i) Its weight = mg (verticaly downwards).
(ii) The tension in the string T along LO.

We can take the horizontal direction at the lowest point ‘p’ as the position of zero gravitational potential energy. Now as per the principle of conservation of energy,

From this relation, we can calculate the tension in the string at the lowest point P, mid-way point and at the highest position of the moving particle.
Case (i) : At the lowest point P, θ = 0°

When the particle completes its motion along the vertical circle, it is referred to as “Looping the Loop” for this the minimum speed at the lowest position must be √5gr

IMPORTANT TABLES

Chapter 4 Motion in a Plane Handwritten Notes Class 11 Physics

Chapter 4 Motion in a Plane Class 11 Notes Physics

Chapter 3 Motion in a Straight Line Handwritten Notes Class 11th Physics

Chapter 3 Motion in a Straight Line Class 11 Notes Physics

Motion In A Straight Line

In this chapter, we shall learn how to describe motion. For this, we develop the concepts of velocity, acceleration and relative velocity. We also develop a set of simple equations called Kinematic equations.

• Motion: Motion is change in position of an object with time.
• Rectilinear motion: The motion along a straight line is called rectilinear motion.
• Point object: If the distance travelled by the body is very large compared with its size, the size of the body may be neglected. The body under such a condition may be taken as a point object. The point object can be represented by a point.

Example:

• The length of bus may be neglected compared with the length of the road it is running.
• The size of planet is ignored compared with the size of the orbit in which it is moving.

Position, Path Length And Displacement

1. Reference point, Frame of reference:
In order to specify position of object, we take reference point and a set of axes. Consider a rectangular coordinate system consisting of three mutually perpendicular axes, labelled x, y, and z axes. The point of intersection of these three axes is called origin (O) and serves as the reference point.

The coordinates (x, y, z) of an object describe the position of the object. To measure time, we place a clock in this coordinate system. This coordinate system along with a clock is called a frame of reference.

To describe the motion along a straight line we can choose x-axis. The position of a carat different time are given in figure 3.1. The position to the right of 0 is taken as positive and to the left of 0 as negative. The position coordinates of point P and Q are +360m +240m. The position coordinate of R is-120m.

2. Path Length (Distance):
The total length of the path travelled by an object is called path length.
Explanation:
Consider a car moving along straight line. The positions of car at different time are given in the x-axis. (See figure 3.1)
Case-1:
The car moves from 0 to P. In this case the distance moved by car is OP = +360.
Case-2:
The car moves from 0 to P and then moves back from P to Q.
In this case, the distance travelled is OP + PQ = +360 + (+120) = +480m.

3. Displacement:
The distance between initial point and final point is called displacement.
OR
The change of position of the particle in a particular direction is called displacement.
Explanation:
Consider a car moving along a straight line. The positions of car at different time is given in the x-axis.
See figure (3.1)
Let us take two cases
Case-1:
The car moves from 0 to P, in this case displacement = (360 – 0) = 360
Case-2:
The car moves from 0 to P and moves back from P to Q.
In this case,
Displacement = 240m
Let x₁ and x₂ be the positions of an object at time t₁ and t₂. Then displacement in time Dt = (t₂ – t₁) can be written as Dx = x₂ – x₁
If x₁ < x₂, Dx is positive and if x₂ < x₁, Dx is negative.
Note: The magnitude of displacement may or may not be equal to the path length traversed by an object.

4. Position Time Graph:
Motion of an object can be represented by a position-time graph.
Position time graph for a stationary object:
For a stationary object, the position does not change with time. Hence the position time graph will be a straight line parallel to time axis.

Question 1.
The position-time of a car is given below. Analyze the graph and explain the motion of car.

Answer:
The car starts from rest a time t=0s from the origin 0 and picks up speed till t=10s. After 10 sec, the car moves with uniform speed till t=18 sec. Then the brakes are applied and the car stops at t = 20s and x = 296m.

Question 2.
Draw the position-time for an object

1. moving with positive velocity
2. moving with negative velocity.

Answer:

Average Velocity And Average Speed

1. Average Velocity:
The average velocity of a particle is the ratio of the total displacement to the time interval.

Explanation:
To explain average velocity, consider a position-time graph of a body given below.

Let x₁ be the position of body at a time t₁ and x₂ be the position at t₂.
The average velocity during the time interval Dt = (t₂ – t₁)

where Dx = x₂ – x₁, and Dt = t₂ – t₁,
¯¯¯vv¯ is the average velocity.

Question 3.
Find the slope of position-time graph given below of uniform motion and explain the result.

Answer:

Slope of displacement time graph gives average velocity.

Question 4.
Displacement time graph of a car is given below.

1. Find the average velocity during the time interval 5 to 7 sec.
2. Find the average velocity by taking slope in the interval 5 to 7 sec.

Answer:
1.

2. Slope, tan q

In this case, slope and average velocity are equal in the same interval.

2. Average Speed:
Average speed of a particle is the ratio of the total distance to total time taken.

Question 5.
A car is moving along a straight line. Say OP in figure. It moves from 0 to P in 18s and returns from P to Q in 6s. What are the average velocity and average speed of the car in going?

1. From 0 to P? and
2. from 0 to P and back to Q. (See Figure 3.1)

Answer:
1. Average velocity

Average speed

In this case the average speed is equal to the magnitude of the average velocity.

2. In this case
Average velocity

Average speed

In this case the average speed is not equal to the magnitude of the average velocity. This happens because the motion here involves change in direction. So that the distance is greater than displacement.
Note: In general, the velocity is always less than or equal to speed.

Instantaneous Velocity And Speed

Nonuniform Motion:
A body is said to be nonuniform motion, if it undergoes unequal displacements in equal intervals of time.

OR

A body moving with varying velocity is called nonuniform motion.

1. Instantaneous Velocity:
Question 6.
Why the concept of instantaneous velocity is introduced?
Answer:
In nonuniform motion the average velocity tells us how fast the object has been moving over a given interval. But it does not tell us how it moves at different instants during that interval. For this we define instantaneous velocity. The velocity at an instant is called instantaneous velocity.
Explanation:
Position-time of a body moving along a straight line is given below.

Let us find average velocity in the interval 2 sec (3s to 5s), centered at t = 4 sec. In this case, the slope of line P₁P₂ give the value of average velocity, ie. Slope of P₁P₂,

Decrease the value of Dt from 2.to 1 sec. (ie. 3.5 to 4.5 sec). Then line P₁P₂ becomes Q₁Q₂. Then the slope of gives average velocity overthe interval 3.5 sec to 4.5sec.
ie. slope of Q₁Q₂

In the limit Dt ® 0, gives the instantaneous velocity at t = 4sec and its value is nearly 3.84m/s.

Question 7.
When average velocity of a body becomes instantaneous velocity?
Answer:
In the limit, Dt goes to zero, the average velocity becomes instantaneous velocity.

But lim lim

\Instantaneous velocity,

Here dx/dt is the differential coefficient of x with respect to time. It is the rate of change of position with respect to time at an instant.

Question 9.
The position of an object moving along x-axis is given by x = a + bt² where a = 8.5m, b = 2.5 m/s² and t is measured in seconds

1. What is the velocity at t = 0s and t = 2s.
2. What is the average velocity between t = 2s and t = 4s?

Answer:
1.

when t = 0
we get v = 2 × 2.5 × 0
v = 0
when t = 2sec
v = 2 × 2.5 × 2 v = 10m/s.

2. The average velocity

Note: If a body is moving with constant velocity, the average velocity is the same as instantaneous velocity at all instants.

2. Instantaneous Speed:
The speed at an instant is called instantaneous speed.
Note:

• The average speed over a finite interval of time is greater or equal to the magnitude of the average velocity.
• Instantaneous speed at an instant is equal to the magnitude of the instantaneous velocity at that instant.

Acceleration

1. Average Acceleration:
Average acceleration of a particle is ratio of the change in velocity to the time interval.

Explanation
Consider a body moving along a straight line. Let v₁ and v₂ be the instantaneous velocities at time t₁ and t₂ respectively.

where Dv = change in velocity, Dt = Time interval

2. Instantaneous Acceleration:
Acceleration at any instant is called instantaneous acceleration.
Explanation
In the limit Dt ® 0, (Dt goes to zero) the average acceleration becomes instantaneous acceleration.
ie. Instantaneous acceleration

Instantaneous acceleration is the rate of change of velocity with respect to time.

3. Uniform Acceleration:
A body is said to be in uniform acceleration if velocity changes equally in equal intervals of time.

Question 10.
The velocities of two bodies A and B are given in the tables. From this table, find which body is moving with uniform acceleration. Explain.

Answer:
Body A is moving with uniform acceleration be-cause the velocity of body increases at the rate of 2 m/s².
The body B is moving with constant velocity. Hence this motion is called uniform motion.

4. Velocity-Time Graph For Uniformly Accelerated Motion:

An example for velocity-time of a uniformly accelerated motion is given in the above figure.
Let v_t1 and v_t2 be the velocities at instants t₁ and t₂respectively.
The slope of graph in the interval (t₂ – t₁) can be written as,

∴ tan q = acceleration
Thus the slope of the velocity-time gives the acceleration of the particle.

Question 11.
The Velocity-time of a body is given below. From this graph draw the corresponding acceleration time graph.

Answer:
The slope of velocity-time graph increases in the interval (0 – 10) sec which means that acceleration of the body increases in this interval.

Velocity is constant in the interval (10 – 18) sec. Hence ’ the slope is zero which means that acceleration is zero in this range.

The slope in the interval (18 – 20) sec is constant and negative. Hence acceleration in this is a negative value. The acceleration – time graph for the above motion is given below.

Question 12.
The position-time graph of a car is given below.

1. Draw corresponding velocity-time graph. Explain the reason for your answer.
2. From velocity-time graph draw acceleration-time graph and identify the regions of

• positive acceleration
• Negative acceleration
• zero acceleration.

Answer:
1. In the time interval (0 – t₁) sec, the slope of x – t graph increases which means that velocity is increasing in this time interval.

In the time interval (t₁ – t₂) sec, slope is constant. Hence velocity remains constant in this time interval.

In the time interval (t₂ – t₃) sec, the slope is decreasing and finally becomes zero. Which means that velocity decreases to zero.

2. Slope is constant throughout the interval (0 – t₁) sec which means that acceleration constant.

In the interval (t₁ – t₂) sec, slope is zero. Which means that acceleration is zero in this region.

Slope is constant (but negative) in the interval (t₂ – t₃)sec. Hence acceleration is constant and negative in this time interval.

Question 13.
Find the region of

1. positive acceleration
2. zero acceleration
3. negative acceleration from the above x-t graph

Answer:

1. Region OA – Positive acceleration
2. Region AB – zero acceleration
3. Region BC – Negative acceleration

Question 14.
Match the following.

Answer:
1) – d, 2) – c, 3) – b, 4) – a.

5. Area Under Velocity-Time Graph:
Area under velocity-time graph represents the displacement over a given time interval.
Explanation
Consider a body moving with constant velocity v. Its velocity-time graph is given below.

Kinematic Equations For Uniformly Accelerated Motion

For uniformly accelerated motion, we can derive some simple equations.

1. Velocity-time relation
2. Position-time relation
3. Position-velocity relation

These equations are called kinematic equations for uniformly accelerated motion.

Consider a body moving along a straight line with uniform acceleration ‘a’. Let ‘u’ be initial velocity and ‘v ‘ be the final velocity at time t.
We know acceleration a =  Change in velocity  Time interval  Change in velocity  Time interval 
a = v−utv−ut
at = v – u

Consider a body moving along a straight line with uniform acceleration a. Let ‘u’ be initial velocity and ‘v’ be the final velocity. ‘S’ is the displacement travelled by the body during the time interval ‘t‘.
Displacement of the body during the time interval t,
S = average velocity × time
S=(v+u2)tS=(v+u2)t _____(1)
But v = u + at ____(2)
Substitute eq.(2) in eq.(1), we get

3. Position-Velocity Relation:
S=(v+u2)tS=(v+u2)t _____(1)
But v = u + at
v−uav−ua = t _____(2)
Substitute eq.(2) in eq.(1)

Free-fall:
An object released (near the surface of earth) is accelerated towards the earth. If air resistance is neglected, the object is said to be in free fall. The acceleration due to gravity near the surface of earth is 9.8 m/s².
Note: Free-fall is a case of motion with uniform acceleration.

Question 15.
A body is allowed to fall freely. Draw the following graph.

1. Acceleration-time
2. Velocity-time
3. Position-time

Answer:
1.

2.

3.

Stopping distance of vehicles:
When brakes are applied to a moving vehicle, the distance it travels before stopping is called stopping distance.

Question 16.
Derive an expression for stopping distance of a vehicle in terms of initial velocity (u) and retardation (a).
Answer:
Let the distance travelled by the vehicle before it stops be ‘s’.
Then we can find ‘s’ using the formula
v² = u² + 2as
0 = u² + -2as

3.7 Relative Velocity
Suppose the distance between two bodies changes with time in magnitude, or in direction or in both. Then each body is said to have a velocity relative to the other.

For example, consider two cars A and B moving in the same direction with equal velocities. To a person in A, the car B would appear to be rest.

Hence the velocity of B relative to A is zero.
ie. V_BA = 0
Similarly, the velocity of A with respect to B is zero.
or V_AB = 0
Let A be moving with a velocity V_A and B be moving with a greater velocity V_B in the same direction. Then the person in A feels that the car B is moving away from him with a velocity V_BA. The velocity of B relative to A

For an observer in B, car A is going back with a velocity. The velocity of A relative to B
V_AB = -(V_B – V_B).

Question 17.
The position-time graph of two bodies A and B (at different situations) are given in the following graphs. Find the relative velocities of the following graph.

Answer:
a) The slope of Aand B are equal. Hence velocity of A and B are equal. So velocity of A with respect to B, V_AB = 0

b) The body A and B meet at t = 3sec

Velocity of A w.r. to B, V_AB = V_A – V_B
= 20-10 = 10 m/s Velocity of B w.r. to A, V_BA = V_B – V_A
= 10 – 20 = -10 m/s

c) The body A and B meet at t = 1 sec.
The velocity of body in the interval t = 1 sec,

Velocity of A w. r. to B,
V_AB = V_A – V_B
= 20 – 10 = 30 m/s
Similarly velocity of B w.r. to A,
V_BA = V_B – V_A
= 10 – +20 = -30 m/s
The magnitude of V_BA or V_AB (=30 m/s) is greater than the magnitude of velocity A or that of B.

Chapter 2 Units and Measurement Handwritten Notes Class 11th Physics

These Notes are Useful for CBSE, MPSB & Other state boards. Textbook with Our notes is sufficient for your XI Examination.

Chapter 2 Units and Measurement Class 11 Notes Physics

Introduction

a. Fundamental or base quantities:
Physics is based on measurement of physical quantities. Certain physical quantities are chosen as fundamental or base quantities. Length, mass, time, electric current thermodynamic temperature, amount of substance and luminous intensity are such base quantities.

b. Units: Fundamental Units and Derived Units Unit:
Measurement of any physical quantity is made by comparing it with a standard. Such standard of measurement are known as unit. If length of rod is 5 m, it means that the length of rod is 5 times the standard unit ‘metre’.

Fundamental Unit:
The unit of fundamental or base quantities are called fundamental or base units. The base units are listed in table.

Base quantity	Base unit
Length	Metre
Mass	kilogram
Time	Second
Electric current	Ampere
Thermodynamic Temperature	Kelvin
Amount of Substance	mole
Luminous Intensity	Candela

Derived Unit

The units of other physical quantities can be expressed as combination of base units. Such units are called derived units.
Example: Unit of force is kgms^-2 (or Newton). Unit of velocity is ms^-1.

The International System Of UnitsDerived Unit
System of Units: A complete set of fundamental and derived units is called a system of unit.

a. Different system of units:
The different systems of units are CGS system FPS (or British) system, MKS system and SI system. A comparison of these systems of unit is given in the table below, (for length, mass and time)

Note: The first three systems of units were used in earlier time. Presently we use SI system.

b. International System Of Unit (Si Unit):
The internationally accepted system of unit for measurement is system international d’ unites (French for International System of Units). It is abbreviated as SI.

The SI system is based on seven fundamental units and these units have well defined and internationally accepted symbols, (given in table – 2.1)

c. Solid Angle and Plane Angle:
Other than the seven base units, two more units are defined.
1. Plane angle (dq): It is defined as ratio of length of arc (ds) to the radius, r.

The unit of plane angle is radian. Its symbol is rad.

2. Solid Angle (dW): It is defined as the ratio of the intercepted area (dA) of spherical surface, to square of its radius.

The unit of solid angle is steradian. The symbol is Sr.

Measurement Of Length

Two methods are used to measure length

• direct method
• indirect method.

The metre scale, Vernier caliper, screwgauge, spherometer are used in direct method for measurement of length. The indirect method is used if range of length is beyond the above ranges.

1. Measurement Of Large Distances:
Parallax Method:
Parallax method is used to find distance of planet or star from earth. The distance between two points of observation (observatories) is called base. The angle between two directions of observation at the two points is called parallax angle or parallactic angle (q).

Parallax Method
The planet ‘s’ is at a distance ‘D’ from the surface of earth. To measure D, the planet is observed from two observatories A and B (on earth). The distance between A and B is b and q be the parallax angle between direction of observation from A and B.

AB can be considered as an arch A h B of length ‘b’ of a circle of radius D with its center at S. (Because q is very small, bDbD<<1], Thus from arch-radius relation.

Thus by measuring b and q distance to planet can be determined. The size of planet or angular diameter of planet can be measured using the value of D. If the angle a (angle between two directions of observation of two diametrically opposite points on planet) is measured using a

Where d is diameter of planet.

2. Estimation Of Very Small Distances:
Size Of Molecule
Electron microscope can measure distance of the order of 0.6A0 (wavelength of electron).

3. Range Of Lengths:
The size of the objects in the universe vary over a very wide range. The table (given below) gives the range and order of lengths and sizes of some objects in the universe.

Units for short and large lengths
1 fermi = 1f = 10^-15m
1 Angstrom = 1A° = 10^-10m
1 astronomical unit = 1AU = 1.496 × 10¹¹m
1 light year = 1/y = 9.46 × 10¹⁵m
(Distance that light travels with velocity of 3 × 10⁸ m/s in 1 year)
1 par sec = 3.08 × 10¹⁶m = 3.3 light year
(par sec is the distance at which average radius of earth’s orbit subtends an angle of 1 arc second).

Measurement Of Mass

Mass is basic property of matter. The S.l. unit of mass is kg. While dealing with atoms and molecules, the kilogram •is an inconvenient unit. In this case there is an important standard unit called the unified atomic mass unit( u).
1 unified atomic mass unit = lu
= (1/12)^th of the mass of carbon-12

1. Range Of Masses:
The masses of the objects in the universe vary over a very wide range which is given in the table.

Measurement Of Time

To measure any time interval we need a clock. We now use an atomic standard of time, which is based on the periodic vibrations produced in a cesium atom. This is the basis of the cesium clock sometimes called atomic clock.

Definition of second:
One second was defined as the duration of 9, 192, 631, 770 internal oscillations between two hyperfine levels of Cesium-133 atom in the ground state.
Range and Order of time intervals

Accuracy, Precision Of Instruments And Errors In Measurement

Error:
The result of every measurement by any measuring instrument contains some uncertainty. This uncertainty is called error.

Systematic errors:
Systematic errors are those errors that tend to be in one direction, either positive or negative.

Sources of systematic errors

1. Instrumental errors
2. Imperfection in experimental technique or procedure
3. personal errors

1. Instrumental errors:
Instrumental error arise from the errors due to imperfect design or calibration of the measuring instrument.
eg: In Vernier Callipers, the zero mark of vernier scale may not coincide with the zero mark of the main scale.

2. Imperfection in experimental technique or procedure:
To determine the temperature of a human body, a thermometer placed under the armpit will always give a temperature lower than the actual value of the body temperature. Other external conditions (such as changes in temperature, humidity, velocity……..etc) during the experiment may affect the measurement.

3. Personal Errors:
Personal error arise due to an individual’s bias, lack of proper setting of the apparatus or individual carelessness etc.

Random errors
The random errors are those errors, which occur irregularly and hence are random with respect to sign and size. These can arise due to random and unpredictable fluctuations in experimental conditions (eg. unpredictable fluctuations in temperature, voltage supply, etc.)

Least Count Error
The smallest value that can be measured by the measuring instrument is called its least count. The least count error is the error associated with the resolution of the instrument. By using instruments of higher precision, improving experimental technique etc, we can reduce least count error.

1. Absolute Error, Relative Error And Percentage Error:
The magnitude of the difference between the true value of the quantity and the measured value is called absolute error in the measurement. Since the true value of the quantity is not known, the arithmetic mean of the measured values may be taken as the true value.

Explanation:
Suppose the values obtained in several measurements are a₁, a₂, a₃,………,a_n. Then arithmetic mean can be written as

The absolute error,
∆a₁ = a_mean – a₁
∆a₂ = a_mean – a₂
∆a_n = a_mean – a_n

a. Mean absolute error:
The arithmetic mean of all the absolute errors is known as mean absolute error. The mean absolute error in the above case,

b. Relative error:
The relative error is the ratio of the mean absolute error (Da_mean) to the mean value (a_mean).

c. Percentage error:
The relative error expressed in percent is called the percentage error (da).

Example:
Question 1.
When the diameter of a wire is measured using a screw gauge, the successive readings are found to be 1.11 mm, 1.14mm, 1.09mm, 1.15 mm, and 1.16 mm. Calculate the absolute error and relative error in the measurement.
Answer:
The arithmetic mean value of the measurement is

The absolute errors in the measurements are
1.13 – 1.14 = 0.02mm
1.13 – 1.14 = -0.01mm
1.13 – 1.09 = 0.04mm
1.13 – 1.15 =-0.02 mm
1.13 – 1.16 = 0.03mm
The arithmetic mean of the absolute errors

Percentage of relative error

2. Combination Of Errors:
When a quantity is determined by combining several measurements, the errors in the different measurements will combine in some way or other.

a. Error of a sum or a difference:
Rule: when two quantities are added or subtracted, the absolute error in the final result is the sum of the absolute errors in the individual quantities.
Explanation:
Let two quantities A and B have measured values A ± DA and B ± DB respectively. DA and DB are the absolute errors in their measurements. To find the error Dz that may occur in the sum z = A + B,
Consider
z + ∆z = (A ± ∆A) + B ± ∆B = (A + B) ± ∆A ± ∆B
The maximum possible error in the value of z is given by,

Similarly, it can be shown that, the maximum error in the difference.
Z = A – B is also given by

b. Error of product ora quotient:
Rule: When two quantities are multiplied or divided, the relative error in the result is the sum of the relative errors in the multipliers.
Explanation:
Suppose Z=AB and the measured values of A and B are A + DA and B + DB. They
Z + DZ = (A + DA) (B + DB)
= AB ± BDA ± ADB ± DADB
Dividing LHS by Z and RHS by AB, we get

c. Errors in case of a measured quantity raised to a power:
Suppose Z = A²

Hence, the relative error in A² is two time the error in A.
In general, if Z=APBqCTZ=APBqCT
Then

Hence the rule: The relative error in a physical quantity raised to the power K is the K times the relative error in the individual quantity.

Significant Figures

Every measurement involves errors. Hence the result of measurement should be reported in a way that indicates the precision of measurement.

Normally, the reported result of measurement is a number that includes all digits in the number that are known reliable plus the first digit that is uncertain. The reliable digits plus the first uncertain digit are known as significant digits or significant figures.
Example:

• The length of a rod measured is 3.52cm. Here there are 3 significant figures. The digits 3 and 5 are reliable and the last digit 2 is uncertain.
• The mass of a body measured as 3.407g. Here there are four significant figures. The figure 7 is uncertain.

When the measurement becomes more accurate, the number of significant figure is increased.
Rules to find significant figures:
1. All the non zero digits are significant.
Example:
Question 1.
Find significant figure of

• 2500
• 263.25

Answer:

• In this case, there are two nonzero numbers. Hence significant figure is 2.
• In this, there are 5 nonzero numbers. Hence significant figure is 5.

2. All the zeros between two nonzero digits are significant, no matter where the decimal point is,
Example:
Question 2.
Find the significant figure

• 2.05
• 302.005
• 2000145

Answer:

• Significant figure is 3
• Significant figure is 6
• Significant figure is 7

3. If the number is less than 1, the zeros on the right of decimal point but to the left to the first nonzero digits are not significant.
Example:
Question 1.
Find the significant figure of

• 0.002308
• 0.000135

Answer:

• 4 significant figures
• 3 significant figures

4. The terminal zeros in a number without a deci¬mal point are not significant.
Example:
Question 1.
Find the significant figure of

• 12300
• 60700

Answer:

• 3
• 3

Note: But if the number obtained is on the basis of actual measurement, all zeros to the right of last non zero digit are significant.
Example: If distance is measured by a scale as 2010m. This contain 4 significant figures.

5. The terminal zeros in a number with a decimal point are significant.
Example:
Question 1.
Find the significant figure of

• 3.500
• 0.06900
• 4.7000

Answer:

• 4
• 4
• 5

Method to find significant figures through scientific notation:
In this notation, every number is expressed as a × 10^b, where a is a number between 1 and 10 and b is any positive or negative power. In this method, we write the decimal after the first digit.
Example:
4700m =4.700 × 10³m
The power of 10 is irrelevant to the determination of significant figures. But all zeros appearing in the base number in the scientific notation are significant. Hence each number in this case has 4 significant figures.
Significant figures in numbers:-

Numbers	Significant figures
1374	4
13.74	4
0.1374	4
0.01374	4
013740	5
1374.0	5
5100	2
51.00	4
5.100	4
3.51 × 103	3
2.1 × 10-2	2
0.4 × 10-4	1

a. Rules for Arithmetic operations with significant figures:
1. Rules for multiplication or division:
In multiplication or division, the computed result should not contain greater number of significant digits than in the observation which has the fewest significant digits.
Examples:
(i) 53 × 2.021 =107.113
The answer is 1.1 × 10² since the number 53 has only 2 significant digits.

(ii) 3700 10.5 = 352.38
The answer is 3.5 × 10² since the minimum number of significant figure is 2 (in the number 3700)

2. Rules for Addition and Subtraction:
In addition or substraction of given numbers, the same number of decimal places is retained in the result as are present in the number with minimum number of decimal places.
Examples:
(i) 76.436 +
12.5
88.936
The answer is 88.9, since only one decimal place is found in the number 12.5.

(ii) 43.6495 +
4.31
47.9595
The answer is 47.96 since only two decimal places are to be retained.

(iii) 8.624 –
3.1726
5.4514
The answer is 5.451

(iv) 6.5 × 10^-5 – 2.3 × 10^-6 = 6.5 × 10^-5 – 0.23 × 10^-5
= 6.27 × 10^-5
The answer is = 6.3 × 10^-5

Dimensions And Dimensional Analysis

All physical quantities can be expressed in terms of seven fundamental quantities. (Mass, length, time, temperature, electric current, luminous intensity and amount of substance). These seven quantities are called the seven dimensions of the physical world.

The dimensions of the three mechanical quantities mass, length and time are denoted by M, L and T. Other dimensions are denoted by K (for temperature), I (for electric current), cd (for luminous intensity) and mol (for the amount of substance).

The letters [L], [M], [T] etc. specify only the nature of the unit and not its magnitude. Since area may be regarded as the product of two lengths, the dimensions of area are represented as [L] × [L] = [L]².

Similarly, volume being the product of three lengths, its dimensions are represented by [L]³. Density being mass per unit volume, its dimensions are M/L³ or M¹L³.

Thus, the dimensions of a physical quantity are the powers to which the fundamental units of length, mass, time must be raised to represent it.
Note: The dimensions of a physical quantity and the dimensions of its unit are the same.

Dimensional Formula And Dimensional Equations

An equation obtained by equating a quantity with its dimensional formula is called dimensional equations of the physical quantities.
Examples:
Consider for example, the dimensions of the following physical quantities.
1. Velocity: Velocity = distance/ time = L/T = L¹T^-1 \The dimension of velocity are, zero in mass, 1 in length and-1 in time.

2. Acceleration:
Acceleration =  Change in velocity  time =L1T−1T=L1T−2 Change in velocity  time =L1T−1T=L1T−2

3. Force: Force = mass × acceleration
Dimensions of force = M × L¹T^-2 = M¹L¹T ^-2
That is, the dimensions of force are 1 in mass, 1 in length and -2 in time.

4. Momentum: Momentum = mass × velocity
Dimensions of momentum = M × L¹T^-1 = M¹L¹T ^-1

5. Moment of a force: Moment = force × distance
Dimensions of moment = M¹L¹T^-2 × L = M¹L²T ^-2

6. Impulse: Impulse = force × time
Dimensions of impulse = M¹L¹T^-2 × T = M¹L¹T ^-1

7. Work: Work = force × distance
Dimensions of work = M¹L¹T^-2 × L = M¹L²T ^-2

8. Energy: Energy = Work done
Dimensions of energy = dimensions of work = M¹L²T^-2.

9. Power: Power = work/time
Dimensions of power =M2L2T−2Tp=M2L2T−2Tp = M¹L²T^-3

Dimensional Analysis And Its Applications

The important uses of dimensional equations are:

1. To check the correctness of an equation.
2. To derive a correct relationship between different physical quantities.
3. To convert one system of units into another.

1. Checking the correctness of an equation:
For the correctness of an equation, the dimensions on either side must be the same. This ‘ is known as the principle of homogeneity of dimensions.

If an equation contains more than two terms, the dimensions of each term must be the same. Thus, if x = y + z, Dimensions of x = dimensions of y = dimensions of z
Example :
Question 1.
Check the correctness of the equation s = ut + 1/2at² by the method of dimensions.
Dimensions of, s = L¹
Dimensions of, u = L¹T^-1
Dimensions of, ut = L¹T^-1 × T¹ = L¹
Dimensions of, a = L¹T^-2
Dimensions of, at² = L¹T^-2 × T² = L¹
The constant 1/2 has no dimensions. Each term has dimension L¹.
Therefore, dimensions of, ut + 1/2 at² = ¹
Thus, either side of the equation has the same dimen¬sion L1 and hence the equation is dimensionally correct.
Note: Even though the equation is dimensionally correct, it does not mean that the equation is necessarily correct. For instance the equation s = ut + at² is also dimensionally correct, though the correct equation, s = ut + 1/2 at².

2. Deriving the correct relationship between different physical quantities:
The principle of homogeneity of dimensions also helps to derive a relationship between the different physical quantities involved. This method is known as dimensional analysis.
Example :
Question 1.
Deduce an expression for the period of oscillation of a simple pendulum.
The period of the simple pendulum may possibly depend upon

• The mass of the bob, m
• The length of the pendulum, I
• Acceleration due to gravity, g
• The angle of swing, q

Let us write the equation for the time period as t = km^a l^b g^c θ^d
where, k is a constant having no dimensions; a, b, c are to be found out. ’
The dimensions of, t = T¹
Dimensions of m = M¹
Dimensions of, l = L¹
Dimensions of, g = L¹T^-2
Angle q has no dimensions (since, q = arc/radius = L/L) Equating the dimensions of both sides of the equation, we get,
T¹ = M^aL^b (L¹T^-2)^c
ie. T¹ = M^aL^b+cT^-2c
The dimensions of the terms on both sides must be the same. Equating the powers of M, L and T.
a = 0; b + c = 0; -2c = 1
∴ c = −12−12, b = ^–c = \(\frac{1}{2}{/latex]
Hence, the equation becomes,
t = kl^1/2, 2g^-1/2
ie, t = k[latex]\sqrt{l/g}\)
Experimentally, the value of k is found to be 2p.
Limitations of Dimensional Analysis:
The method of dimensional analysis has the following limitations:

• It gives no information about the dimensionless constant involved in the equation.
• The method is not applicable to equations involving trigonometric and exponential functions.
• This method cannot be employed to derive the exact form of the relationship, if it contains sum
  of two, or more terms.
• If the given physical quantity depends on more than three unknown quantities, the method fails.

3. Conversion of one system of units to another:
Suppose we have a physical quantity of dimensions a, b and c in mass, length and time. The dimensional formula for the quantity is therefore, M^aL^bT^c. Let its numerical value be n, in one system in which the fundamental units of mass, length and time are M₁, L₁ and T₁ respectively. Then, the magnitude of the physical quantity
= n₁ M₁^aL₁^bT₁^c
Also, let the numerical value of the same quantity be n₂ in another system where the fundamental units of mass, length and time are M₂, L₂ and T₂respectively. Then the magnitude of the quantity
= n₂ M₂^aL₂^bT₂^c
Equating, n₂ M₂^aL₂^bT₂^c =
n₁ M₁^aL₁^bT₁^c

Example :
Question 1.
Find the number of dynes in one newton.
Answer:
Dyne is the unit of force in the C.G.S. system and newton is the S.I.unit. The dimensional formula for force is M¹L¹T^-2. In eqn. (1) let the suffix 1 refer to quantities in S.I and 2 those in the C.G.S. system.
Here, a = 1, b = 1 and c = 2

and n₁ = 1 (ie. one Newton)
By eqn. (1),
n₂ = 1 (1000)¹ (100)¹ (1)^-2 = 10⁵
ie. 1 newton = 10⁵ dynes.