Class 11 physics key notes Part 1

Physics - Meaning

The term derived from Greek word meaning nature. Sanskrit equivalent of the word Physics is Bhautiki that is used to refer to the study of the physical world.
In a broader sense Physics is the study of the basic laws of nature and their manifestation. Physics is all about explaining diverse physical phenomena with the help of few concepts and laws.

Scope and Excitement of Physics

In Physics, there are two domains of interest macroscopic and microscopic.
Macroscopic domain It includes phenomena at the laboratory, terrestrial and astronomical scales.
Microscopic domain It includes atomic, molecular and nuclear phenomena.
However, recently a third domain of interest between macroscopic domain and microscopic domain (Mesoscopic) has also come in light. In this domain scientists deals with a few tens or hundreds of atoms, has emerged as an exciting field of research.

Various theories related to macroscopic domain and microscopic domain are further categorized as:

Classical Physics

It is the study of macroscopic phenomena. It includes subjects as
Mechanics Under mechanics, we study:
i. Newton’s laws of motion
ii. The law of gravitation is concerned with the motion (or equilibrium) of particles, rigid and deformable bodies, and general systems of particles.)
Electrodynamics It deals with electric and magnetic phenomena associated with charged and magnetic bodies.
Thermodynamics It deals with systems in macroscopic equilibrium and is concerned with changes in internal energy, temperature, entropy, etc. of the system through external work and transfer of heat, the efficiency of heat engines and refrigerators etc.
OpticsIt is the study of phenomenon connected with light and optical instruments like telescope, microscope etc.

Quantum Theory

It is the framework for explaining microscopic phenomena as classical physics can’t explain phenomenon at microscopic level (or smaller dimensions like atoms, nuclei etc.)

Physics, Technology and Society

There are number of examples in the world which shows close relation between physics, technology and society. Such as, the steam engine is inseparable from the Industrial Revolution in England in the 18th century, which had great impact on the course of human civilization. Wireless communication technology, computers are some other examples.

Some physicists from different countries of the world and their major contributions

Name	Major Contribution /Discovery	Country of Origin
Archimedes	Principle of buoyancy; Principle of the lever	Greece
Galileo Galilei	Law of inertia	Italy
Christiaan Huygens	Wave theory of light	Holland
Isaac Newton	Universal law of gravitation; Laws of motion;Reflecting telescope	U.K.
Michael Faraday	Laws of electromagnetic induction	U.K.
James Clerk Maxwell	Electromagnetic theory; Light-an electromagnetic wave	U.K.
Heinrich Rudolf Hertz	Generation of electromagnetic waves	Germany
J.C. Bose	Ultra short radio waves	India
W.K. Roentgen	X-rays	Germany
J.J. Thomson	Electron	U.K.
Marie Sklodowska Curie	Discovery of radium and polonium; Studies on natural radioactivity	Poland
Albert Einstein	Explanation of photoelectric effect; Theory of relativity	Germany
Victor Francis Hess	Cosmic radiation	Austria
R.A. Millikan	Measurement of electronic charge	U.S.A.
Ernest Rutherford	Nuclear model of atom	New Zealand
Niels Bohr	Quantum model of hydrogen atom	Denmark
C.V. Raman	Inelastic scattering of light by molecules	India
Louis Victor de Borglie	Wave nature of matter	France
M.N. Saha	Thermal ionisation	India
S.N. Bose	Quantum statistics	India
Wolfgang Pauli	Exclusion principle	Austria
Enrico Fermi	Controlled nuclear fission	Italy
Werner Heisenberg	Quantum mechanics; Uncertainty principle	Germany
Paul Dirac	Relativistic theory of electron; Quantum statistics	U.K.
Edwin Hubble	Expanding universe	U.S.A.
Ernest Orlando Lawrence	Cyclotron	U.S.A.
James Chadwick	Neutron	U.K.
Hideki Yukawa	Theory of nuclear forces	Japan
Homi Jehangir Bhabha	Cascade process of cosmic radiation	India
Lev Davidovich Landau	Theory of condensed matter; Liquid helium	Russia
S. Chandrasekhar	Chandrasekhar limit, structure and evolution of stars	Russia
John Bardeen	Transistors; Theory of super conductivity	U.S.A.
C.H. Townes	Maser; Laser	U.S.A.
Abdus Salam	Unification of weak and electromagnetic interactions	Pakistan

Link between technology and physics

Technology	Scientific principle(s)
Steam engine	Laws of thermodynamics
Nuclear reactor	Controlled nuclear fission
Radio and Television	Generation, propagation and detection of electromagnetic waves
Computers	Digital logic
Lasers	Light amplification by stimulated emission of radiation
Production of ultra high magnetic fields	Superconductivity
Rocket propulsion	Newton’s laws of motion
Electric generator	Faraday’s laws of electromagnetic induction
Aeroplane	Bernoulli’s principle in fluid dynamics
Particle accelerators	Motion of charged particles in electromagnetic fields
Sonar	Reflection of ultrasonic waves
Optical fibres	Total internal reflection of light
Non-reflecting coatings	Thin film optical interference
Electron microscope	Wave nature of electrons
Photocell	Photoelectric effect
Fusion test reactor (Tokamak)	Magnetic confinement of plasma
Giant Metrewave Radio Telescope (GMRT)	Detection of cosmic radio waves
Bose-Einstein condensate	Trapping and cooling of atoms by laser beams and magnetic fields.

Fundamental Forces in Nature

Four fundamental forces in nature that govern the diverse phenomena of the macroscopic and the microscopic world are given below
a)Gravitational Force
b)Electromagnetic Force
c)Strong Nuclear Force
d)Weak Nuclear Force

Basic Properties of Fundamental Forces in Nature

Name	Relative Strength (& Range)	Operates among
Gravitational force	10-39(Infinite)	All objects in the universe
Weak nuclear force	10-13(Very short, Sub-nuclear size: ∼10–16m)	Some elementary particles, particularly electron and neutrino
Electromagnetic force	10-2 (Infinite)	Charged particles
Strong nuclear force	1 (Short, nuclear size 10-15m	Nucleons, heavier elementary particles

Conservation Laws in Physics

The physical quantities that remain unchanged in a process are called conserved quantities. Some of the general conservation laws in nature include the laws of conservation of energy, mass, linear momentum, angular momentum, charge, parity, etc. Some conservation laws are true for one fundamental force but not for the other

Class 11 Physics Notes CBSE NCERT New Edition Based Unit 10 : Hydrostatics

Class 11 Physics Notes

👇

1. Unit 1 : Measurement
2. Unit 2 : Vectors
3. Unit 3 : Motion in Straight Line
4. Unit 4 : Projectile Motion & Circular Motion
5. Unit 5 : Laws of Motion
6. Unit 6 : Work, Power & Energy
7. Unit 7 : Rotational Motion
8. Unit 8 : Gravitation
9. Unit 9 : Elasticity
10. Unit 10 : Hydrostatics
11. Unit 11 : Hydrodynamics
12. Unit 12 : Surface Tension
13. Unit 13 : Thermometry and Calorimetry
14. Unit 14 : Kinetic Theory of Gases
15. Unit 15 : Thermodynmics
16. Unit 16 : Transmission of Heat
17. Unit 17 : Oscillations
18. Unit 18 : Waves & Sound

Unit 10 : Hydrostatics

MECHANICAL PROPERTIES OF FLUIDS

Fluids

Fluids are those substances which can flow when an external force is applied on it.
Liquids and gases are fluids.
Fluids do not have finite shape but takes the shape of the containing vessel.
The total normal force exerted by liquid at rest on a given surface is called thrust of liquid.
The SI unit of thrust is newton.
In fluid mechanics the following properties of fluid would be considered
(i) When the fluid is at rest - hydrostatics
(ii) When the fluid is in motion - hydrodynamics

Pressure Exerted by the Liquid

The normal force exerted by a liquid per unit area of the surface in contact is called pressure of liquid orhydrostatic pressure.

Pressure exerted by a liquid column

p = hρg

Where, h = height of liquid column, ρ = density of liquid

and g = acceleration due to gravity

Mean pressure on the walls of a vessel containing liquid upto height h is (hρg / 2).

Pascal’s Law

The increase in pressure at a point in the enclosed liquid in equilibrium is transmitted equally in all directions in liquid and to the Walls of the container.
The working of hydraulic lift, hydraulic press and hydraulic brakes are based on Pascal’s law.

Atmospheric Pressure

The pressure exerted by the atmosphere on earth is atmospheric pressure.
It is about 100000 N/m².
It is equivalent to a weight of 10 tones on 1 m².
At sea level, atmospheric pressure is equal to 76 cm of mercury column. Then, atmospheric pressure
= hdg = 76 x 13.6 x 980 dyne/cm²
[The atmospheric pressure does not crush our body because the pressure of the blood flowing through our circulatory system] balanced this pressure.]
Atmospheric pressure is also measured in torr and bar.
1 torr = 1 mm of mercury column
1 bar = l0⁵ Pa
Aneroid barometer is used to measure atmospheric pressure.

Buoyancy

When a body is partially or fully immersed in a fluid an upward force acts on it, which is called buoyant force or simply buoyancy.
The buoyant force acts at the centre of gravity of the liquid displaced] by the immersed part of the body and this point is called the centre buoyancy.

Archimedes’ Principle

When a body is partially or fully immersed in a liquid, it loses some of its weight. and it is equal to the weight of the liquid displaced by the immersed part of the body.
If T is the observed weight of a body of density σ when it is fully immersed in a liquid of density p, then real weight of the body
w = T / ( 1 – p / σ)

Laws of Floatation

A body will float in a liquid, if the weight of the body is equal to the weight of the liquid displaced by the immersed part of the body.
If W is the weight of the body and w is the buoyant force, then
(a) If W > w, then body will sink to the bottom of the liquid.
(b) If W < w, then body will float partially submerged in the liquid.
(c) If W = w, then body will float in liquid if its whole volume is just immersed in the liquid,
The floating body will be in stable equilibrium if meta-centre (centre of buoyancy) lies vertically above the centre of gravity of the body.
The floating body will be in unstable equilibrium if meta-centre (centre of buoyancy) lies vertically below the centre of gravity of the body.
The floating body will be in neutral equilibrium if meta-centre (centre of buoyancy) coincides with the centre of gravity of the body.

Density and Relative Density

Density of a substance is defined as the ratio of its mass to its volume.
Density of a liquid = Mass / Volume
Density of water = 1 g/cm³ or l0³ kg/m³
It is scalar quantity and its dimensional formula is [ML^-3].
Relative density of a substance is defined as the ratio of its density to the density of water at 4°C,
Relative density = Density of substance / Density of water at 4°C
= Weight of substance in air / Loss of weight in water
Relative density also known as specific gravity has no unit, no dimensions.
For a solid body, density of body = density of substance
While for a hollow body, density of body is lesser than that of Substance.
When immiscible liquids of different densities are poured in a container, the liquid of highest density will be at the bottom while, that of lowest density at the top and interfaces will be plane.

Density of a Mixture of Substances

When two liquids of mass m1 and m2 having density p1 and p2 are mixed together then density of mixture is
p = m1 + m2 / (m1 /p1 ) + (m2 + p2)
= p1p2 (m1 + m2) / (m1p2 + m2p1)
When two liquids of same mass m but of different densities p1 and p2 are mixed together then density of mixture is
p = 2p1p2 / p1 + p2
When two liquids of same volume V but of different densities p1 and p2 are mixed together then density of mixture is
p = p1 + p2 / 2
Density of a liquid varies with pressure
p = po [ 1 + Δp / K]
where, po = initial density of the liquid, K = bulk modulus of elasticity of the liquid and Δp = change in pressure

Class 11 Physics Notes CBSE NCERT New Edition Based Unit 9 : Elasticity

Class 11 Physics Notes

👇

1. Unit 1 : Measurement
2. Unit 2 : Vectors
3. Unit 3 : Motion in Straight Line
4. Unit 4 : Projectile Motion & Circular Motion
5. Unit 5 : Laws of Motion
6. Unit 6 : Work, Power & Energy
7. Unit 7 : Rotational Motion
8. Unit 8 : Gravitation
9. Unit 9 : Elasticity
10. Unit 10 : Hydrostatics
11. Unit 11 : Hydrodynamics
12. Unit 12 : Surface Tension
13. Unit 13 : Thermometry and Calorimetry
14. Unit 14 : Kinetic Theory of Gases
15. Unit 15 : Thermodynmics
16. Unit 16 : Transmission of Heat
17. Unit 17 : Oscillations
18. Unit 18 : Waves & Sound

Unit 9 : Elasticity

MECHANICAL PROPERTIES OF SOLIDS

Deforming Force

A force which produces a change in configuration of the object on applying it, is called a deforming force.

Elasticity

Elasticity is that property of the object by virtue of which it regain its original configuration after the removal of the deforming force.

Elastic Limit

Elastic limit is the upper limit of deforming force upto which, if deforming force is removed, the body regains its original form completely and beyond which if deforming force is increased the body loses its property of elasticity and get permanently deformed.

Perfectly Elastic Bodies

Those bodies which regain its original configuration immediately and completely after the removal of deforming force are called perfectly elastic bodies. e.g., quartz and phosphor bronze etc.

Perfectly Plastic Bodies

Those bodies which does not regain its original configuration at all on the removal of deforming force are called perfectly plastic bodies, e.g., putty, paraffin, wax etc.

Stress

The internal restoring force acting per unit area of a deformed body is called stress.
Stress = Restoring force / Area
Its unit is N/m² or Pascal and dimensional formula is [ML^-12T^-2].
Stress is a tensor quantity.

Stress is of Two Types

(i) Normal Stress If deforming force is applied normal to the area, then the stress is called normal stress.
If there is an increase in length, then stress is called tensile stress.
If there is a decrease in length, then stress is called compression stress.
(ii) Tangential Stress If deforming force is applied tangentially, then the stress is called tangential stress.

Strain

The fractional change in configuration is called strain.
Strain = Change in the configuration / Original configuration
It has no unit and it is a dimensionless quantity.
According to the change in configuration, the strain is of three types
(1) Longitudinal strain= Change in length / Original length
(2) Volumetric strain = Change in volume / Original volume
(iii) Shearing strain = Angular displacement of the plane perpendicular to the fixed surface.

Hooke’s Law

Within the limit of elasticity, the stress is proportional to the strain.
Stress infinity Strain
or Stress = E * Strain
where, E is the modulus of elasticity of the material of the body.

Types of Modulus of Elasticity

1. Young’s Modulus of Elasticity
It is defined as the ratio of normal stress to the longitudinal strain Within the elastic limit.
y = Normal stress / Longitudinal strain
y = FΔl / Al = Mg Δl / πr²l Its unit is N/m² or Pascal and its dimensional formula is [ML^-1T^-2].
2. Bulk Modulus of Elasticity
It is defined as the ratio of normal stress to the volumetric strain within the elastic limit.
K = Normal stress / Volumetric strain
K = FV / A ΔV = Δp V / Δ V
where, Δp = F / A = Change in pressure.
Its unit is N/m2 or Pascal and its dimensional formula is [ML^-1T^-2].
3. Modulus of Rigidity (η)
It is defined as the ratio of tangential stress to the shearing strain, within the elastic limit.
η = Tangential stress / Shearing strain
Its unit is N/m² or Pascal and its dimensional formula is [ML^-1T^-2].

Compressibility

Compressibility of a material is the reciprocal of its bulk modulus of elasticity.
Compressibility (C) = 1 / k
Its SI unit is N^-1m² and CGS unit is dyne^-1 cm².
Steel is more elastic than rubber. Solids are more elastic and gases are least elastic.
For liquids. modulus of rigidity is zero.
Young’s modulus (Y) and modulus of rigidity (η) are possessed by solid materials only.

Limit of Elasticity

The maximum value of deforming force for which elasticity is present in the body is called its limit of elasticity.

Breaking Stress

The minimum value of stress required to break a wire, is called breaking stress.
Breaking stress is fixed for a material but breaking force varies with area of cross-section of the wire.
Safety factor = Breaking stress / Working stress

Elastic Relaxation Time

The time delay in restoring the original configuration after removal of deforming force is called elastic relaxation time.
For quartz and phosphor bronze this time is negligible.

Elastic After Effect

The temporary delay in regaining the original configuration by the elastic body after the removal of deforming force, is called elastic after effect.

Elastic Fatigue

The property of an elastic body by virtue of which its behaviour becomes less elastic under the action of repeated alternating deforming force is called elastic fatigue.

Ductile Materials

The materials which show large plastic range beyond elastic limit are called ductile materials, e.g., copper, silver, iron, aluminum, etc.
Ductile materials are used for making springs and sheets.

Brittle Materials

The materials which show very small plastic range beyond elastic limit are called brittle materials, e.g., glass, cast iron, etc.

Elastomers

The materials for which strain produced is much larger than the stress applied, with in the limit of elasticity are called elastomers, e.g., rubber, the elastic tissue of aorta, the large vessel carrying blood from heart. etc.
Elastomers have no plastic range.

Elastic Potential Energy in a Stretched Wire

The work done in stretching a wire is stored in form of potential energy of the wire.
Potential energy U = Average force * Increase in length
= 1 / 2 FΔl
= 1 / 2 Stress * Strain * Volume of the wire
Elastic potential energy per unit volume
U = 1 / 2 * Stress * Strain
= 1 / 2 (Young’s modulus) * (Strain)²
Elastic potential energy of a stretched spring = 1 / 2 kx²
where, k = Force constant of spring and x = Change in length.

Thermal Stress

When temperature of a rod fixed at its both ends is changed, then the produced stress is called thermal stress.
Thermal stress = F / A = yαΔθ
where, α = coefficient of linear expansion of the material of the rod.
When temperature of a gas enclosed in a vessel is changed, then the thermal stress produced is equal to change in pressure (Δp)of the gas.
Thermal stress = Δ p = Ky Δ θ
where, K = bulk modulus of elasticity and
γ = coefficient of cubical expansion of the gas.
Interatomic force constant
K = Yr_o
where, r_o = interatomic distance.

Poisson’s Ratio

When a deforming force is applied at the free end of a suspended wire of length 1 and radius R, then its length increases by dl but its radius decreases by dR. Now two types of strains are produced by a single force.
(i) Longitudinal strain = ΔU l
(ii) Lateral strain = – Δ R/ R
∴ Poisson’s Ratio (σ) = Lateral strain / Longitudinal strain = – Δ R/ R / ΔU l
The theoretical value of Poisson’s ratio lies between – 1 and 0.5. Its practical value lies between 0 and 0.5.

Relation Between Y, K, η and σ

(i) Y = 3K (1 – 2σ)
(ii) Y = 2 η ( 1 + σ)
(iii) σ = 3K – 2η / 2η + 6K
(iv) 9 / Y = 1 / K + 3 / η or Y = 9K η / η + 3K

Important Points

Coefficient of elasticity depends upon the material, its temperature and purity but not on stress or strain.
1. For the same material, the three coefficients of elasticity γ, η and K have different magnitudes.
2. Isothermal elasticity of a gas ET = ρ where, ρ = pressure of the gas.
3. Adiabatic elasticity of a gas Es = γρ
where, γ = Cp / Cv ratio of specific heats at constant pressure and at constant volume.
4. Ratio between isothermal elasticity and adiabatic elasticity Es/ ET = γ = Cp / Cv

Cantilever

A beam clamped at one end and loaded at free end is called a cantilever.
Depression at the free end of a cantilever is given by
δ = wl³ / 3YI_G

where, w = load, 1 = length of the cantilever,
y = Young’s modulus of elasticity, and IG = geometrical moment of inertia.
For a beam of rectangular cross-section having breadth b and thickness d.
I_G = bd³ / 12
For a beam of circular cross-section area having radius r,
I_G = π r⁴ / 4

Beam Supported at Two Ends and Loaded at the Middle

Depression at middle δ = wl³ / 48YI_G

Torsion of a Cylinder

where, η = modulus of rigidity of the material of cylinder,
r = radius of cylinder,
and 1 = length of cylinder,
Work done in twisting the cylinder through an angle θ
W = 1 / 2 Cθ²
Relation between angle of twist (θ) and angle of shear (φ)
rθ = lφ or φ = r / l = θ

Class 11 Physics Notes CBSE NCERT New Edition Based Unit 8 : Gravitation

Class 11 Physics Notes

👇

1. Unit 1 : Measurement
2. Unit 2 : Vectors
3. Unit 3 : Motion in Straight Line
4. Unit 4 : Projectile Motion & Circular Motion
5. Unit 5 : Laws of Motion
6. Unit 6 : Work, Power & Energy
7. Unit 7 : Rotational Motion
8. Unit 8 : Gravitation
9. Unit 9 : Elasticity
10. Unit 10 : Hydrostatics
11. Unit 11 : Hydrodynamics
12. Unit 12 : Surface Tension
13. Unit 13 : Thermometry and Calorimetry
14. Unit 14 : Kinetic Theory of Gases
15. Unit 15 : Thermodynmics
16. Unit 16 : Transmission of Heat
17. Unit 17 : Oscillations
18. Unit 18 : Waves & Sound

Unit 8 : Gravitation

GRAVITATION

Every object in the universe attracts every other object with a force which is called the force of gravitation.

Gravitation is one of the four classes of interactions found in nature.

These are

(i) the gravitational force

(ii) the electromagnetic force

(iii) the strong nuclear force (also called the hadronic force).

(iv) the weak nuclear forces.

Although, of negligible importance in the interactions of elementary particles, gravity is of primary importance in the interactions of objects. It is gravity that holds the universe together.

Newton’s Law of Gravitation

Gravitational force is a attractive force between two masses m1 and m2 separated by a distance r.

The gravitational force acting between two point objects is proportional to the product of their masses and inversely proportional to the square of the distance between them.

Gravitational force(F)=Gm₁m₂/r²

where G is universal gravitational constant.

The value of G is 6.67 X 10^-11 Nm² kg^-2 and is same throughout the universe.

The value of G is independent of the nature and size of the bodies well as the nature of the medium between them.

Dimensional formula of Gis [M^-1L³T^-2].

Important Points about Gravitation Force

(i) Gravitational force is a central as well as conservative force.

(ii) It is the weakest force in nature.

(iii) It is 1036 times smaller than electrostatic force and 10’l8times smaller than nuclear force.

(iv) The law of gravitational is applicable for all bodies, irrespective of their size, shape and position.

(v) Gravitational force acting between sun and planet provide it centripetal force for orbital motion.

(vi) Gravitational pull of the earth is called gravity.

(vii) Newton’s third law of motion holds good for the force of gravitation. It means the gravitation forces between two bodies are action-reaction pairs.

Following three points are important regarding the gravitational force

(i) Unlike the electrostatic force, it is independent of the medium between the particles.

(ii) It is conservative in nature.

(iii) It expresses the force between two point masses (of negligible volume). However, for external points of spherical bodies the whole mass can be assumed to be concentrated at its centre of mass.

Note Newton’s law of gravitation holde goods for object lying at uery large distances and also at very short distances. It fails when the distance between the objects is less than 10-9 m i.e., of the order of intermolecular distances.

Acceleration Due to Gravity

The uniform acceleration produced in a freely falling object due to the gravitational pull of the earth is known as acceleration due to gravity.

It is denoted by g and its unit is m/s². It is a vector quantity and its direction is towards the centre of the earth.

The value of g is independent of the mass of the object which is falling freely under gravity.

The value of g changes slightly from place to place. The value of g is taken to be 9.8 m/s² for all practical purposes.

The value of acceleration due to gravity on the moon is about. one sixth of that On the earth and on the sun is about 27 times of that on the earth.

Among the planets, the acceleration due to gravity is minimum on the mercury.

Relation between g and a is given by

g = Gm / R²

where M = mass of the earth = 6.0 * 10²⁴ kg and R = radius of the earth = 6.38 * 10⁶ m.

Acceleration due to gravity at a height h above the surface of the earth is given by

g^h = Gm / (R+h)² = g (1 – 2h / R)

Factors Affecting Acceleration Due to Gravity

(i) Shape of Earth Acceleration due to gravity g 1 / R² Earth is elliptical in shape. Its diameter at poles is approximately 42 km less than its diameter at equator.

Therefore, g is minimum at equator and maximum at poles.

(ii) Rotation of Earth about Its Own Axis If ω is the angular velocity of rotation of earth about its own axis, then acceleration due to gravity at a place having latitude λ is given by

g’ = g – Rω² cos² λ

At poles λ = 90° and g’ = g

Therefore, there is no effect of rotation of earth about its own axis at poles.

At equator λ = 0° and g’ = g – Rω²

The value of g is minimum at equator

If earth stapes its rotation about its own axis, then g will remain unchanged at poles but increases by Rω² at equator.

(iii) Effect of Altitude The value of g at height h from earth’s surface

g’ = g / (1 + h / R)²

Therefore g decreases with altitude.

(iv) Effect of Depth The value of gat depth h A from earth’s surface

g’ = g * (1 – h / R)

Therefore g decreases with depth from earth’s surface.

The value of g becomes zero at earth’s centre.

Gravitational Field

The space in the surrounding of any body in which its gravitational pull can be experienced by other bodies is called gravitational field.

Intensity of Gravitational Field

The gravitational force acting per unit mass at Earth any point in gravitational field is called intensity of gravitational field at that point.

It is denoted by E^g or I.

E^g or I = F / m

Intensity of gravitational field at a distance r from a body of mass M is given by

E^g or I = GM / r²

It is a vector quantity and its direction is towards the centre of gravity of the body.

Its S1 unit is N/m and its dimensional formula is [LT^-2].

Gravitational mass M_g is defined by Newton’s law of gravitation.

M_g = F_g / g = W / g = Weight of body / Acceleration due to gravity

∴ (M₁)g / (M₂)g = F_g1g2 / F_g2g1

Gravitational Potential

Gravitational potential at any point in gravitational field is equal the work done per unit mass in bringing a very light body from infinity to that point.

It is denoted by V_g.

Gravitational potential, V_g = W / m = – GM / r

Its SI unit is J / kg and it is a scalar quantity. Its dimensional formula is [L³r^-2].

Since work W is obtained, that is, it is negative, the gravitational potential is always negative.

Gravitational Potential Energy

Gravitational potential energy of any object at any point in gravitational field is equal to the work done in bringing it from infinity to that point. It is denoted by U.

Gravitational potential energy U = – GMm / r

The negative sign shows that the gravitational potential energy decreases with increase in distance.

Gravitational potential energy at height h from surface of earth

U_h = – GMm / R + h = mgR / 1 + h/R

Satellite

A heavenly object which revolves around a planet is called a satellite. Natural satellites are those heavenly objects which are not man made and revolve around the earth. Artificial satellites are those neaven objects which are man made and launched for some purposes revolve around the earth.
Time period of satellite
T = 2π √r³ / GM
= 2π √(R + h)³ / g [ g = GM / R²
Near the earth surface, time period of the satellite
T = 2π √R³ / GM = √3π / Gp
T = 2π √R / g = 5.08 * 10³ s = 84 min.
where p is the average density of earth.

Artificial satellites are of two types

1. Geostationary or Parking Satellites

A satellite which appears to be at a fixed position at a definite height to an observer on earth is called geostationary or parking satellite.

Height from earth’s surface = 36000 km

Radius of orbit = 42400 km

Time period = 24 h

Orbital velocity = 3.1 km/s

Angular velocity = 2π / 24 = π / 12 rad / h

There satellites revolve around the earth in equatorial orbits.

The angular velocity of the satellite is same in magnitude and direction as that of angular velocity of the earth about its own axis.

These satellites are used in communication purpose.

2. Polar Satellites

These are those satellites which revolve in polar orbits around earth. A polar orbit is that orbit whose angle of inclination with equatorial plane of earth is 90°. 

Height from earth’s surface = 880 km

Time period = 84 min

Orbital velocity = 8 km / s

Angular velocity = 2π / 84 = π / 42 rad / min.

There satellites revolve around the earth in polar orbits.

These satellites are used in forecasting weather, studying the upper region of the atmosphere, in mapping, etc.

Orbital Velocity

Orbital velocity of a satellite is the minimum velocity required to the satellite into a given orbit around earth.

Orbital velocity of a satellite is given by

v_o = √GM / r = R √g / R + h where, M = mass of the planet, R = radius of the planet and h = height of the satellite from planet’s surface.

If satellite is revolving near the earth’s surface, then r = (R + h) =- R

Now orbital velocity,

v_o = √gR

= 7.92km / h

if v is the speed of a satellite in its orbit and vo is the required orbital velocity to move in the orbit, then

(i) If v < vo, then satellite will move on a parabolic path and satellite falls back to earth.

(ii) If V = vo then satellite revolves in circular path/orbit around earth.

(iii) If vo < V < ve then satellite shall revolve around earth in elliptical orbit.

Energy of a Satellite in Orbit

Total energy of a satellite

E = KE + PE

= GMm / 2r + (- GMm / r)

= – GMm / 2r

Binding Energy

The energy required to remove a satellite from its orbit around the earth (planet) to infinity is called binding energy of the satellite.

Binding energy of the satellite of mass m is given by

BE = + GMm / 2r

Escape Velocity

Escape velocity on earth is the minimum velocity with which a body has to be projected vertically upwards from the earth’s surface so that it just crosses the earth’s gravitational field and never returns.

Escape velocity of any object

v_e = √2GM / R

= √2gR = √8πp GR² / 3

Escape velocity does not depend upon the mass or shape or size of the body as well as the direction of projection of the body.

Escape velocity at earth is 11.2 km / s.

Some Important Escape Velocities

Moon - 2.3 km/s

Mercury - 4.28 km/s

Earth - 11.2 km/s

Jupiter - 60 km/s

Sun - 618 km/s

Neutron star - 2 x 10⁵ km/s

Escape velocity

Relation between escape velocity and orbital velocity of the satellite

v_e = √2 vo

If velocity of projection U is equal the escape velocity (v = ve), then the satellite will escape away following a parabolic path.

If velocity of projection u of satellite is greater than the escape velocity ( v > v_e), then the satellite will escape away following a hyperbolic path.

Weightlessness

It is a situation in which the effective weight of the body becomes zero.
Weightlessness is achieved
(i) during freely falling under gravity
(ii) inside a space craft or satellite
(iii) at the centre of the earth
(iv) when a body is lying in a freely falling lift.

Kepler’s Laws of Planetary Motion

(i) Law of orbit Every planet revolve around the sun in elliptical orbit and sun is at its one focus.

(ii) Law of area The radius vector drawn from the sun to a planet sweeps out equal areas in equal intervals of time, i.e., the areal velocity of the planet around the sun is constant.

Areal velocity of a planet

dA / dt = L / 2m = constant 

(iii) Law of period The square of the time period of revolution of planet around the sun is directly proportional to the cube semi-major axis of its elliptical orbit.

Important Points

(i) A missile is launched with a velocity less than the escape velocity. The sum of its kinetic energy and potential energy is negative.
(ii) The orbital speed of jupiter is less than the orbital speed of earth.
(iii) A bomb explodes on the moon. You cannot hear the sound of the explosion on earth.
(iv) A bottle filled with water at 30°C and fitted with a cork is taken to the moon. If the cork is opened at the surface of the moon then water will boil.
(v) For a satellite orbiting near earth’s surface

(a) Orbital velocity = 8 km / s

(b) Time period = 84 min approximately

(c) Angular speed ω = 2π / 84 rad / min

= 0.00125 rad / s

(vi) Inertial mass and gravitational mass

(a) Inertial mass = force / acceleration

(b) Gravitational mass = weight of body / acceleration due to gravity

(c) They are equal to each other in magnitude.

(d) Gravitational mass of a body is affected by the presence of other bodies near it. Inertial mass of a body remains unaffected by the presence of other bodies near it.

Class 11 Physics Notes CBSE NCERT New Edition Based Unit 7 : Rotational Motion

Class 11 Physics Notes

👇

1. Unit 1 : Measurement
2. Unit 2 : Vectors
3. Unit 3 : Motion in Straight Line
4. Unit 4 : Projectile Motion & Circular Motion
5. Unit 5 : Laws of Motion
6. Unit 6 : Work, Power & Energy
7. Unit 7 : Rotational Motion
8. Unit 8 : Gravitation
9. Unit 9 : Elasticity
10. Unit 10 : Hydrostatics
11. Unit 11 : Hydrodynamics
12. Unit 12 : Surface Tension
13. Unit 13 : Thermometry and Calorimetry
14. Unit 14 : Kinetic Theory of Gases
15. Unit 15 : Thermodynmics
16. Unit 16 : Transmission of Heat
17. Unit 17 : Oscillations
18. Unit 18 : Waves & Sound

Unit 7 : Rotational Motion

SYSTEM OF PARTICLES AND ROTATIONAL MOTION

Centre of Mass

Centre of mass of a system is the point that behaves as whole mass of the system is concentrated at it and all external forces are acting on it.

For rigid bodies, centre of mass is independent of the state of the body i.e., whether it is in rest or in accelerated motion centre of mass will rermain same.

Centre of Mass of System of n Particles

If a system consists of n particles of masses m1, m2, m3 ,… mn having position vectors rl, r2, r3,… rn. then position vector of centre of mass of

Centre of Mass of Two Particle System

Choosing O as origin of the coordinate axis.
1.Then, position of centre of mass from m₁=m₂d/(m1+m2)
(ii) Position of centre of mass from m2 = (m₁d) / m1 + m2
iii) If position vectors of particles of masses m1 and m2 are r1 and r2respectively, then

(iv) If in a two particle system, particles of masses m1 and m2 moving with velocities v1 and v2 respectively, then velocity the centre of mass

(v) If accelerations of the particles are a1, and a1respectively, then acceleration of the centre of mass

(vi) Centre of mass of an isolated system has a constant velocity.
(vii) It means isolated system will remain at rest if it is initially rest or will move with a same velocity if it is in motion initially.
(viii) The position of centre of mass depends upon the shape, size and distribution of the mass of the body.
(ix) The centre of mass of an object need not to lie with in the object.
(x) In symmetrical bodies having homogeneous distribution mass the centre of mass coincides with the geometrical centre the body.
(xi) The position of centre of mass of an object changes translatory motion but remains unchanged in rotatory motion,

Translational Motion

A rigid body performs a pure translational motion, if each particle the body undergoes the same displacement in the same direction in given interval of time.

Rotational Motion

A rigid body performs a pure rotational motion, if each particle of the body moves in a circle, and the centre of all the circles lie on a straight line called the axes of rotation.

Rigid Body

If the relative distance between the particles of a system do not changes on applying force, then it called a rigtd body. General motion of a rigid body consists of both the translational motion and the rotational motion.

Moment of Inertia

The inertia of rotational motion is called moment of inertia. It is denoted by L.
Moment of inertia is the property of an object by virtue of which it opposes any change in its state of rotation about an axis.
The moment of inertia of a body about a given axis is equal to the sum of the products of the masses of its constituent particles and the square of their respective distances from the axis of rotation.

Its unit is kg.m2 and its dimensional formula is [ML²].
The moment of inertia of a body depends upon
1. position of the axis of rotation
2. orientation of the axis of rotation
3. shape and size of the body
4. distribution of mass of the body about the axis of rotation.
The physical significance of the moment of inertia is same in rotational motion as the mass in linear motion.

The Radius of Gyration

The root mean square distance of its constituent particles from the axis of rotation is called the radius of gyration of a body.
It is denoted by K.
Radius of gyration

The product of the mass of the body (M) and square of its radius gyration (K) gives the same moment of inertia of the body about rotational axis.
Therefore, moment of inertia I = MK² ⇒ K = √1/M

Parallel Axes Theorem

The moment of inertia of any object about any arbitrary axes is equal to the sum of moment of inertia about a parallel axis passing through the centre of mass and the product of mass of the body and the square of the perpendicular distance between the two axes.
Mathematically I = I_CM + Mr²
where I is the moment of inertia about the arbitrary axis, IcM is moment of inertia about the parallel axis through the centre of mass, M is the total mass of the object and r is the perpendicular distance between the axis.

Perpendicular Axes Theorem

The moment of inertia of any two dimensional body about an axis perpendicular to its plane (Iz) is equal to the sum of moments of inertia of the body about two mutually perpendicular axes lying in its own plane and intersecting each other at a point, where the perpendicular axis passes through it.
Mathematically I_z = I_x + I_y
where I_xand I_y are the moments of inertia of plane lamina about perpendicular axes X and Y respectively which lie in the plane lamina an intersect each other.
Theorem of parallel axes is applicable for any type of rigid body whether it is a two dimensional or three dimensional, while the theorem of perpendicular is applicable for laminar type or two I dimensional bodies only.

Moment of Inertia of Homogeneous Rigid Bodies

For a Circular Disc

For a Thin Rod

For a Solid Cylinder

For a Rectangular Plate

For a Thin Spherical Shell

For a Solid Sphere

Equations of Rotational Motion

(i) ω = ω0 + αt
(ii) θ = ω0t + 1/2 αt2
(iii) ω2 = ω02 + 2αθ
where θ is displacement in rotational motion, ω0 is initial velocity, omega; is final velocity and a is acceleration.

Torque

Torque or moment of a force about the axis of rotation
τ = r x F = rF sinθ n
It is a vector quantity.
If the nature of the force is to rotate the object clockwise, then torque is called negative and if rotate the object anticlockwise, then it is called positive. Its SI unit is ‘newton-metre’ and its dimension is [ML²T^-2].
In rotational motion, torque, τ = Iα
where a is angular acceleration and 1is moment of inertia.

Angular Momentum

The moment of linear momentum is called angular momentum.
It is denoted by L.
Angular momentum, L = I ω = mvr
In vector form, L = I ω = r x mv
Its unit is ‘joule-second’ and its dimensional formula is [ML²T^-1].
Torque, τ = dL/dt

Conservation of Angular Momentum

If the external torque acting on a system is zero, then its angular momentum remains conserved.
If τ_ext0, then L = I(ω) = constant ⇒ I1ω1== I2ω2

Angular Impulse

Total effect of a torque applied on a rotating body in a given time is called angular impulse. Angular impulse is equal to total change in angular momentum of the system in given time.
Thus, angular impulse