Topics
Rotational Dynamics
 Rotational Dynamics
 Characteristics of Circular Motion
 Applications of Uniform Circular Motion
 Vertical Circular Motion
 Moment of Inertia as an Analogous Quantity for Mass
 Radius of Gyration
 Theorems of Perpendicular and Parallel Axes
 Torque and Angular Momentum
 Conservation of Angular Momentum
 Rolling Motion
Circular Motion
 Angular Displacement
 Angular Velocity
 Angular Acceleration
 Angular Velocity and Its Relation with Linear Velocity
 Uniform Circular Motion
 Radial Acceleration
 Dynamics of Uniform Circular Motion  Centripetal Force
 Centrifugal Forces
 Banking of Roads
 Vertical Circular Motion Due to Earth’s Gravitation
 Equation for Velocity and Energy at Different Positions of Vertical Circular Motion
 Kinematical Equations for Circular Motion in Analogy with Linear Motion.
Gravitation
 Newton’s Law of Gravitation
 Projection of Satellite
 Periodic Time
 Kepler’S Laws
 Binding Energy and Escape Velocity of a Satellite
 Weightlessness
 Variation of ‘G’ Due to Lattitude and Motion
 Acceleration Due to Gravity and Its Variation with Altitude and Depth
 Communication satellite and its uses
 Composition of Two S.H.M.’S Having Same Period and Along Same Line
Mechanical Properties of Fluids
Angular Momentum
Kinetic Theory of Gases and Radiation
 Kinetic Theory of Gases and Radiation
 Behaviour of a Gas
 Ideal Gas and Real Gas
 Mean Free Path
 The Pressure of Ideal Gas
 Root Mean Square (RMS) Speed
 Kinetic Interpretation of Temperature
 Law of Equipartition of Energy
 Specific Heat Capacity
 Absorption, Reflection, and Transmission of Heat Radiation
 Perfect Blackbody
 Emission of Heat Radiation
 Kirchhoff’S Law of Heat Radiation and Its Theoretical Proof
 Spectral Distribution of Blackbody Radiation
 Stefanboltzmann Law of Radiation
Oscillations
 Periodic and Oscillatory Motions
 Simple Harmonic Motion
 Differential Equation of Linear S.H.M.
 Projection of U.C.M.(Uniform Circular Motion) on Any Diameter
 Phase of K.E (Kinetic Energy)
 K.E.(Kinetic Energy) and P.E.(Potential Energy) in S.H.M.
 Composition of Two S.H.M.’S Having Same Period and Along Same Line
 Some Systems Executing Simple Harmonic Motion
Thermodynamics
Oscillations
 Oscillations
 Explanation of Periodic Motion
 Linear Simple Harmonic Motion (S.H.M.)
 Differential Equation of Linear S.H.M.
 Acceleration (A), Velocity (V) and Displacement (X) of S.H.M.
 Amplitude(A), Period(T) and Frequency (N) of S.H.M.
 Reference Circle Method
 Phase in S.H.M.
 Graphical Representation of S.H.M.
 Composition of Two S.H.M.’S Having Same Period and Along Same Line
 The Energy of a Particle Performing S.H.M.
 Simple Pendulum
 Angular S.H.M. and It's Differential Equation
 Damped Oscillations
 Free Oscillations, Forced Oscillations and Resonance Oscillations
 Periodic and Oscillatory Motions
Elasticity
Surface Tension
Superposition of Waves
Wave Motion
Wave Optics
Stationary Waves
Electrostatics
 Electrostatics
 Application of Gauss' Law
 Electric Potential and Potential Energy
 Electric Potential Due to a Point Charge, a Dipole and a System of Charges
 Equipotential Surfaces
 Electrical Energy of Two Point Charges and of a Dipole in an Electrostatic Field
 Conductors and Insulators, Free Charges and Bound Charges Inside a Conductor
 Dielectrics and Electric Polarisation
 Capacitors and Capacitance, Combination of Capacitors in Series and Parallel
 Displacement Current
 Energy Stored in a Capacitor
 Van De Graaff Generator
 Uniformly Charged Infinite Plane Sheet and Uniformly Charged Thin Spherical Shell (Field Inside and Outside)
Current Electricity
Kinetic Theory of Gases and Radiation
 Concept of an Ideal Gas
 Kinetic Theory of Gases Assumptions
 Mean Free Path
 Derivation for Pressure of a Gas
 Degrees of Freedom
 Derivation of Boyle’s Law
 Thermal Equilibrium
 First Law of Thermodynamics
 Heat Engines
 Temperature and Heat
 Qualitative Ideas of Blackbody Radiation
 Wein'S Displacement Law
 Green House Effect
 Stefan's Law
 Maxwell Distribution
 Specific Heat Capacities  Gases
 Law of Equipartition of Energy
Magnetic Fields Due to Electric Current
 Magnetic Fields Due to Electric Current
 Magnetic Force
 Cyclotron Motion
 Helical Motion
 Magnetic Force on a Wire Carrying a Current
 Force on a Closed Circuit in a Magnetic Field
 Torque on a Current Loop in Magnetic Field
 Magnetic Dipole Moment
 Magnetic Potential Energy of a Dipole
 Magnetic Field Due to a Current: Biotsavart Law
 Force of Attraction Between Two Long Parallel Wires
 Magnetic Field Produced by a Current in a Circular Arc of a Wire
 Axial Magnetic Field Produced by Current in a Circular Loop
 Magnetic Lines for a Current Loop
 Ampere's Law
 Magnetic Field of a Solenoid and a Toroid
Wave Theory of Light
Magnetic Materials
Interference and Diffraction
 Interference of Light
 Conditions for Producing Steady Interference Pattern
 Interference of Light Waves and Young’S Experiment
 Analytical Treatment of Interference Bands
 Measurement of Wavelength by Biprism Experiment
 Fraunhofer Diffraction Due to a Single Slit
 Rayleigh’s Criterion
 Resolving Power of a Microscope and Telescope
 Difference Between Interference and Diffraction
Electromagnetic Induction
 Electromagnetic Induction
 Faraday's Laws of Electromagnetic Induction
 Lenz's Law
 Flux of the Field
 Motional Electromotive Force
 Induced Emf in a Stationary Coil in a Changing Magnetic Field
 Generators
 Back Emf and Back Torque
 Induction and Energy Transfer
 Eddy Currents
 SelfInductance
 Energy Stored in a Magnetic Field
 Energy Density of a Magnetic Field
 Mutual Inductance
 Transformers
Electrostatics
 Applications of Gauss’s Law
 Mechanical Force on Unit Area of a Charged Conductor
 Energy Density of a Medium
 Dielectrics and Polarisation
 Concept of Condenser
 The Parallel Plate Capacitor
 Capacity of Parallel Plate Condenser
 Effect of Dielectric on Capacity
 Energy of Charged Condenser
 Condensers in Series and Parallel,
 VandeGraaff Generator
AC Circuits
Current Electricity
Dual Nature of Radiation and Matter
Magnetic Effects of Electric Current
Structure of Atoms and Nuclei
Magnetism
Semiconductor Devices
Electromagnetic Inductions
 Electromagnetic Induction
 Faraday’s Law of Induction
 SelfInductance
 Mutual Inductance
 Transformers
 Need for Displacement Current
 Coil Rotating in Uniform Magnetic Induction
 Alternating Currents
 Reactance and Impedance
 LC Oscillations
 Inductance and Capacitance
 Resonant Circuit
 Power in Ac Circuit: the Power Factor
 Lenz’S Law and Conservation of Energy
Electrons and Photons
Atoms, Molecules and Nuclei
 Alphaparticle Scattering and Rutherford’S Nuclear Model of Atom
 Bohr'S Model for Hydrogen Atom
 Hydrogen Spectrum
 Atomic Masses and Composition of Nucleus
 Introduction of Radioactivity
 Law of Radioactive Decay
 MassEnergy Relation and Mass Defect
 Nuclear Binding Energy
 Nuclear Fusion – Energy Generation in Stars
 deBroglie Relation
 Wave Nature of Matter
 Wavelength of an Electron
 DavissonGermer Experiment
 Continuous and Characteristics Xrays
Semiconductors
 Energy Bands in Solids
 Extrinsic Semiconductor
 Applications of Ntype and Ptype Semiconductors
 Special Purpose Pn Junction Diodes
 Semiconductor Diode
 Zener Diode as a Voltage Regulator
 IV Characteristics of Led
 Transistor and Characteristics of a Transistor
 Transistor as an Amplifier (Ceconfiguration)
 Transistor as a Switch
 Oscillators
 Digital Electronics and Logic Gates
Communication Systems
 Elements of a Communication System
 Basic Terminology Used in Electronic Communication Systems
 Bandwidth of Signals
 Bandwidth of Transmission Medium
 Need for Modulation and Demodulation
 Production and Detection of an Amplitude Modulated Wave
 Space Communication
 Propagation of Electromagnetic Waves
 Modulation and Its Necessity
description
 Bernoulli's Theorem and Its Applications
 Bernoulli’s equation
 Speed of Efflux: Torricelli’s Law
 Venturimeter
 Blood Flow and Heart Attack
 Dynamic Lift  Ball moving without spin, Ball moving with spin, Magnus effect, Aerofoil or lift on aircraft wing
notes
Bernoulli’s Principle
For a streamlined fluid flow, the sum of the pressure (P), the kinetic energy per unit volume `((ρv)^2/2)` and the potential energy per unit volume (ρgh) remain constant.
Mathematically: `"P"+ (ρv^2)/2 + ρgh = "constant"`

where P= pressure,

`"E"/ ("Volume")=1/2"mv"^2/V = 1/2v2(m/V) = 1/2ρv2`

`"E"/("Volume") = "mgh"/"V" = ("m"/"V")"gh" = "ρgh"'`
Derive: Bernoulli’s equation:
Mathematically:
Consider the fluid initially lying between B and D. In an infinitesimal time interval `Delta"t"`, this fluid would have moved.
Suppose `v_1`= speed at B and `v_2`= speed at D, initial distance moved by fluid from B to `"C" = "v"_1Delta"t" `
in the same interval `Delta"t"`fluid distance moved by D to `E = v_2Deltat.`
P1 = pressure A1, P2 = pressure at A2
work done on the fluid at left end(BC) `"W"_1 = "P"_1"A"_1("v"_1Delta"t")`
work done by the fluid at the other end (DE)`"W"_2="P"_2"A"_2("v"_2Delta"t")`
Net work done on the fluid is `"W"_1 "W"_2=("P"_1"A"_1"v"_1Delta"t""P"_2"A"_2"v"_2Delta"t")`
By the equation of contunity Av = constant
`"P"_1"A"_1"v"_1Delta"t""P"_2"A"_2"v"_2Delta"t"` where `"A"_1"v"_1Delta"t"="P"_1Delta"V" and "A"_2"v"_2Delta"t" = "P"_2Delta"V"`
Therefore work done = `("P"_1"P"_2) Delta"Vequation"("a")`
Part of this work goes in changing kinetic energy, `Delta"K"=1/2"m"("v"_2^2"v"_1^2)` and part in gravitational potential energy, `Delta"U"="mg"("h"_2"h"_1)`
The total change in energy `Delta"E" = Delta"K" + Delta"U"=1/2"m"("v"_2^2"v"_1^2)+"mg"("h"_2"h"_1)` ...(i)
Density of fluid `rho="m"/"V" or "m"=rho"V"`
Therefore in small interval of time `Delta"t"`,small change in mass `Delta"m"`. `Delta"m"=rhoDelta"V"` ...(ii)
Substituting eq(ii) in eq(i)
`Delta"E"=1/2rhoDelta"V"("v"_2^2"v"_1^2)+rho"g"Delta"V"("h"_2"h"_1)` ....(b)
By using work energy theorem: `"W" = Delta"E"`
From (a) and (b)`
`("P"_1"P"_2)Delta"V"=1/2rhoDelta"V"("v"_2^2"v"_1^2)+rhoDelta"V"("h"_2"h"_1)`
`"P"_1"P"_2 = 1/2rhov_2^21/2rhov_1^2+rho"gh"_2rho"gh"_1` ...(By cancelling `Delta"V"` from both the sides)
After rearranging we get `P_1+1/2rho"v"_1^2 + rho"gh"_1 = 1/2rho"v"_2^2 + rho"gh"_2`
`"P" + 1/2rho"v"^2+rho"gh"="constant"`
This is the bernoullis equation.
The flow of an ideal fluid in a pipe of varying crosssection. The fluid in a section of length `"v"_1`Δt moves to the section of length `"v"2`Δt in time Δt.
Special case:
When a fluid is at rest. this means `v_1=v_2=0`
from Bernoulli's equation `P_1+1/2rho"v"_1^2 + rho"gh"_1 = 1/2rho"v"_2^2 + rho"gh"_2`
By putting `v_1=v_2=0` in the above eq.
`"P"_1"P"_2=rho"g"("h"_2"h"_1)` this eq is same as when the fluids are at rest.
when the pipe is horizontal `h_1=h_2` this means there is no potential energy by the virtue of height.
Therefore from Bernoullis equation `(P_1+1/2rhov_1^2+rhogh_1 = 1/2rhov_2^2+rhogh_2)`
By simplifying, `"P"+1/2rhov^2="constant"`
Torricelli’s law

Torricelli law states that the speed of flow of fluid from an orifice is equal to the speed that it would attain if falling freely for a distance equal to the height of the free surface of the liquid above the orifice.

Consider any vessel which has an orifice (slit)filled with some fluid.

The fluid will start flowing through the slit and according to Torricelli law, the speed with which the fluid will flow is equal to the speed with which a freely falling body attains such that the height from which the body falls is equal to the height of the slit from the free surface of the fluid.

Let the distance between the free surface and the slit = h

The velocity with which the fluid flows is equal to the velocity with which a freely falling body attains if it is falling from a height h.
Derivation:
 Let A_{1}= area of the slit (it is very small), v_{1}= Velocity with which fluid is flowing out.
 A_{2}=Area of the free surface of the fluid,v_{2}=velocity of the fluid at the free surface.
 From Equation of Continuity, Av=constant.Therefore A_{1}v_{1} = A_{2}v_{2}.
 From the figure, A_{2}>>>A_{1}, This implies v_{2}<<v_{1}(This means fluid is at rest on the free surface), Therefore v_{2}~ 0.
 Using Bernoulli’s equation
`"P" + 1/2rho"v"^2+rho"gh"="constant"`
Applying Bernoulli’s equation at the slit:
`"P"_"a"+1/2rho"v"_1^2+rho"gy"_1` ...(1)
where `"P"_a = "Atmospheric pressure", "y"_1="height of the slit from the base".`
applying Bernoullis equation at the surface:
`P+rhogy_2` ...(2)
where as `v_2=0 therefore1/2rhov_1^2=0,y_2="height of the free surface from the base"`
by equation (1) & (2)
`"P"_a + 1/2rho"v"_1^2 + rho"gy"_1 = "P"+rho"gy"_2`
`1/2rhov_1^2=(PP_a) + rhog(y_2y_1)`
`=(PP_a)rhogh("where h"=(y_2y_1))`
`v_1^2=2/rho[(PP_a)+rhogh]`
Therefore `v_1=sqrt2/rho[(PP_a)+rhogh]` this is the velocity by which the fluid will come out of the small slit.
`v_1`is known as speed of Efflux. this means the speed of the fluid outflow.
Torricelli’s law: The speed of efflux, v_{1}, from the side of the container is given by the application of Bernoulli’s equation.
Case1: The vessel is not closed it is open to the atmosphere that means P=Pa.

Therefore `"v"_1=sqrt(2"gh")`.This is the speed of a freely falling body.

This is in accordance with Torricelli’s law which states that the speed by which the fluid is flowing out of a small slit of a container is the same as the velocity of a freely falling body.

Case2:Tank is not open to the atmosphere but P>>P_{a}.

Therefore 2gh is ignored as it is very very large, hence `"v"_1=sqrt(2"p")/rho`

The velocity with which the fluid will come out of the container is determined by the pressure at the free surface of the fluid alone.
Problem: Calculate the velocity of the emergence of a liquid from a hole in the side of a wide cell 15cm below the liquid surface?
Answer:
By using Torricelli’s law `"v"_1=sqrt(2"gh")`
`=2xx9.8xx15xx10^2"m"//"s"`
=1.7m/s
Venturimeter

Venturimeter is a device to measure the flow of incompressible liquid.

It consists of a tube with a broad diameter having a larger crosssectional area but there is a small constriction in the middle.

It is attached to the Utube manometer. One end of the manometer is connected to the constriction and the other end is connected to the broader end of the Venturimeter.

The Utube is filled with a fluid whose density is ρ.

`"A"_1`= crosssectional area at the broader end, `"v"_1` = velocity of the fluid.

`"A"_2`=crosssectional area at the constriction, `"v"2`= velocity of the fluid.

By the equation of continuity, wherever the area is more velocity is less and viceversa. As `"A"_1` is more this implies `"v"_1` is less and viceversa.

The pressure is inversely ∝ to Therefore at `"A"_1` pressure `"P"_1` is less as compared to pressure `"P"_2` at `"A"_2`.

This implies `"P"_1<"P"_2` as `"v"_1>"v"_2`.

As there is a difference in the pressure the fluid moves, this movement of the fluid is marked by the level of the fluid increase at one end of the Utube.
Venturimeter: determining the fluid speed.
By equation of continuity: `"A"_1"v"_1="A"_2"v"_2`
This implies `"v"_2=("A"_1/"A"_2)"V"_1` ...(1)
By bernoullis equation:
`"P"_1+1/2rho"v"_1^2 + rho"gh"_1 = 1/2rho"v"_2^2 + rho"gh"_2`
As height is the same we can ignore the term `rhog`
This implies `P_1P_2=1/2rho(v_2^2v_1^2)`
`=1/2rho(A_1^2/A_2^2(v_1^2v_1^2))`
`=1/2rho(A_1^2/A_2^21)`
As there is pressure difference the level of the fluid in the Utube changes.
`("P"_1"P"_2)=hrho_mg "where" rho_m` (density of the fluid inside the manometer).
`1/2rho"v"_1^2("A"_1^2/"A"_2^21)="h"rho_m"g"`
`v_1=2"h"rho_m"g"/rho["A"_1^2/"A"_2^21]^(1/2)`
Practical Application of Venturimeter:

Spray Gun or perfume bottle They are based on the principle of Venturimeter.

Consider a bottle filled with fluid which has a pipe that goes straight till constriction. There is a narrow end of a pipe that has a greater crosssectional area.

The crosssectional area of constriction which is in the middle is less.

There is pressure difference when we spray as a result some air goes in, the velocity of the air changes depending on the crosssectional area.

Also because of the difference in the crosssectional area there is a pressure difference, the level of the fluid rises and it comes out.

Dynamic Lift

Dynamic lift is the normal force that acts on a body by virtue of its motion through a fluid.

Consider an object which is moving through the fluid, and due to the motion of the object through the fluid, there is a normal force that acts on the body.

This force is known as dynamic lift.

Dynamic lift is most popularly observed in airplanes.

Whenever an aeroplane is flying in the air, due to its motion through the fluid here fluid is air in the atmosphere. Due to its motion through this fluid, there is a normal force that acts on the body in the vertically upward direction.

This force is known as Dynamic lift.

Examples:

Airplane wings

Spinning ball in the air
Dynamic lift on airplane wings:

Consider an aeroplane whose body is streamlined. Below the wings of the aeroplane there is air that exerts an upward force on the wings. As a result, aeroplane experiences a dynamic lift.
Magnus Effect

Dynamic lift by virtue of spinning is known as the Magnus effect.

Magnus effect is a special name given to dynamic lift by virtue of spinning.

Example:Spinning of a ball.
Case1:When the ball is not spinning.

The ball moves in the air it does not spin, the velocity of the ball above and below the ball is the same.

As a result, there is no pressure difference.(ΔP= 0).

Therefore there is no dynamic lift.
Case2: When the ball is moving in the air as well as spinning.

When the ball spins it drags the air above it, therefore, the velocity above the ball is more as compared to the velocity below the ball.

As a result, there is a pressure difference; the pressure is more below the ball.

Because of the pressure difference, there is an upward force which is the dynamic lift.
Problem: A fully loaded Boeing aircraft has a mass of 3.3x10^{5}kg. Its total wing area is 500m^{2}. It is a level flight with a speed of 960km/h. Estimate the pressure difference between the lower and upper surfaces of the wings.
Answer:
Weight of the aircraft= Dynamic lift
`"mg" = ("P"_1"P"_2)"A"`
`"mg"/"A"=DeltaP`
`DeltaP=3.3xx10^5xx9.8/500`
=6.5x10^{3 }N/m^{2}