January 31, 2016Introduction1 Welcome to Subject No. – PH11001 Subject Name – Physics 1 Credit - 4 ( ) (Lecture-Tutorial-Practical-Total) Dr. Shivakiran Bhaktha B.N. Department of Physics

January 31, 2016Introduction2 You can contact me at: o My office : First Floor of Physics Dept (C-220) o My contact Phone Number : o My o For any discussion I am available on Thursdays 5-6 PM

January 31, 2016Introduction3 Course Breakup o 3 Lecture Classes per week Class Room: F 142 (Raman Auditorium) (Sections 6 and 7) Monday: 10:30 AM; Tuesday: 08:30 AM; Thursday:11:30 AM. o 1 Tutorial Class per week (Check your groups) Tutorials start from Tuesday(7th Jan, 2014) More Details Displayed on 1st Year Notice Board in Department of Physics

January 31, 2016Introduction4 Marks Breakup o Mid Semester Exam : 30% o End Semester Exam : 50% o Tutorial : 20% Mid Sem. Exams: 17 th Feb 2014 to 25 th Feb 2014 End Sem. Exams: 21 st Apr 2014 to 29 th Apr 2014

January 31, 2016Introduction5 Course Contents 1.Overview of vibrations (mechanical, electrical, optical). 2.Free, damped, forced oscillations. 3.Coupled oscillations. 4.Wave motion. 5.Electromagnetic waves. 6.Radiation. 7.Optical phenomena (interference, diffraction). 8.Wave mechanics – failure of classical oscillators, quantum oscillators.

January 31, 2016Introduction6 References - Books 1.“Lecture Notes & Problems bank for Physics” by R.S. Saraswat and G.P. Sastry (Available at THACKERS Book shop at Tech. Market) 2.“The Physics of Vibrations and Waves” by H.J. Pain ( Wiley) 3.“Feynman Lectures on Physics: Vol- I” 4.“Optics” by Eugene Hecht 5. PPT slides: Will be made available at regular intervals. Please note: For completeness, you MUST consult books and take Slides as reference. Note: If any other source is used then it would be told in the class

Acknowledgements ©A. K. Das ©A. Dhar ©P. Khastgir ©S. Bharadwaj ©A. Roy ©P. Roy Chaudhuri ©S. Kar ©S. Majumder ©A. Chandra ©S.K. Varshney January 31, 20167SHO

January 31, 2016Introduction8 No use of mobile phones in the class

January 31, 2016Introduction9 Lets begin...

January 31, 2016Introduction10 Chapter 1 Oscillations

January 31, 2016Introduction11 Oscillations in nature are ubiquitous (found everywhere) Some oscillations are visible; some are subtle but it is difficult to find something which never exhibits oscillations. Examples: –A mosquito’s wings, for example, vibrate hundreds of times per second and produce an audible note. –The whole earth, after being jolted by an earthquake, may continue to vibrate at the rate of about one oscillation per hour. –The human body itself is a treasure-house of vibratory phenomena. “After all, our hearts beat, our lungs oscillate, we can hear and speak because our eardrums and larynx vibrate. The light waves which permit us to see entail vibration. We move by oscillating our legs, even the atoms of which we are constituted vibrate…” Importance of Chapter 1: Oscillations

January 31, 2016Introduction12 As a result… ¤ They have an enormous impact on understanding how things work. ¤ In astrophysics, thermal physics, quantum mechanics, optics, condensed-matter physics, mechanics, atmospheric and planetary physics, etc. “So it’s basic literacy in physics”

January 31, 2016Introduction13 Examples Obvious Oscillations »water waves »pendulums »earthquakes »car springs, shock absorbers Less-Obvious Oscillations »musical instruments - guitar, piano, flute »suspension bridges »lasers »quartz-crystal electronic watches »radio antenna »fiber optics

January 31, 2016Introduction14 Subtle Oscillations »heat in a solid »structure of an atom »superconductivity »heat radiation More examples

January 31, 2016Introduction15 Similarly….. “Waves are everywhere” Mechanical waves (sound, water, phonons…)Mechanical waves (sound, water, phonons…) Electromagnetic waves (radiation, visible light…)Electromagnetic waves (radiation, visible light…) Matter waves (atoms)Matter waves (atoms) Gravitational waves (neutron stars, black holes..)Gravitational waves (neutron stars, black holes..)

January 31, 2016Introduction16 Before starting... Lets learn some important notations

January 31, 2016Introduction17 Notations

January 31, 2016Introduction18 Complex Numbers We will extensively use complex numbers throughout this course.We will extensively use complex numbers throughout this course. Their use is NOT mandatory.Their use is NOT mandatory. But the use gives tremendouse convenience in classical wave physics.But the use gives tremendouse convenience in classical wave physics.

January 31, 2016Introduction19 Complex Numbers Definition : And also satisfied by -i

January 31, 2016Introduction20 Complex Numbers x & y : real numbers i : unit imaginary number iy : pure imaginary number

January 31, 2016Introduction21 Imaginary Exponent e i ϕ = cos ϕ + i sin ϕ The combination of exponential series with the complex number notation i is very convenient in physics. Mathematically it is convenient to express sine or cosine oscillatory behaviour in the form of e ix.

January 31, 2016Introduction22 Representation of a complex number in terms of real and imaginary components Im Complex Plane r sin  r cos  Re z  ^ Argand/Phasor Diagram  = tan -1 (b/a) b a z = a + i b: Cartesian representation z = r e i  : Polar representation

January 31, 2016Introduction23 In general Complex conjugate

January 31, 2016Introduction24 An Argand diagram is a plot of complex numbers as points z=x+iy in the complex plane using the x-axis as the real axis and y-axis as the imaginary axis. In the plot the circle represents the complex modulus |z| of z and the angle represents its complex argument.

January 31, 2016Introduction25 Phasor = Rotating Arrow + Associated Phase AnglePhasor The complex argument is also called the phase.

SIMPLE HARMONIC OSCILLATION 26

OSCILLATION Free OSCILLATION Forced OSCILLATION Under NO damping (Undamped Oscilation) Under damping (Damped Oscillation) 27

A simple harmonic oscillator is an oscillating system which satisfies the following properties. 1. Motion is about an equilibrium position at which point no net force acts on the system. 2. The restoring force is proportional to and oppositely directed to the displacement. 3. Motion is periodic. Free Oscillation SHO 28

oneed inertia, or its equivalent mass, for linear motion moment of inertia, for rotational motion inductance, e.g., for electrical circuit oneed a displacement, or its equivalent amplitude (position, voltage, pressure, etc.) oneed a negative feedback to counter inertia displacement-dependent restoring force: spring, gravity, etc. electrical potential restoring charges Elements of an Oscillator 29

Model System 30

Hooke’s Law: Equation of SHM Angular frequency Time period 31

Note: Small angle approximation is valid till ~ 0.4 radians (= 23  ) l Example 32

Torsional Oscillation Where, I = Moment of Inertia θ  = Angular displacement  = Restoring couple Example 33

Example 34

Let‘s find the general solution... The equation of motion is given by: This is a second order linear homogeneous equation with constant coefficients. The general solution is given by: The constansts c 1 and c 2 can be determined by the initial conditions. Free Oscillation SHO 35

Special cases The mass is pulled to one side and released from rest at t=0 The mass is hit and is given a speed v 0 at its equilibrium position at t=0 The mass is given a speed v at a displacement a at t=0 36

Finding Solution A=Amplitude, =Phase Equation of SHM 37

© SB Complex Representation 38

The real part of the complex number Represents the oscillating quantity © SBMeaning/Significance 39

© SB Complex Velocity Note: i acts as an operator 40

Velocity © SB 41 A=2 units

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© Hecht x=ASin  0 t x=Asin(  0 t+  /3) x=Asin(  0 t+  /2) tt tt tt SHM Example What does moving along a circular path have to do with moving back & forth in a straight line (oscillation about equilibrium) ?? 43

x = A cos  = A cos (  t ), since  =  t y x -A A 0  A  8 7 x Simple harmonic motion along x x component of uniform circular motion 44

Harmonic Oscillator Potential The potential energy is found by summing all the small elements of work: (kx.dx). Energy of SHO 45

Because, Energy of SHO 46

Energy of SHO = Sum of potential and kinetic energy remains a constant. Assumption: Ideal case, total energy remains constant. All P.E. becomes K.E. and vice versa. PotentialKinetic 47

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= Time Average Energy of SHO HOW??? 49

Time Average REFRESH 50

Average of Oscillations 51

Time Average Note: Here ‘T’ is the total time of observation. 52

= Time Average Energy of SHO

Root Mean Square (RMS) 54

In case of a sinusoidal wave, the RMS value is easy to calculate. If we define I p to be the amplitude, then: where t is time and ω is the angular frequency (ω = 2π/T, where T is the period of the wave). Since I p is a positive constant: Using a trigonometric identity: but since the interval is a whole number of complete cycles (per definition of RMS), the sin terms will cancel out, leaving: You are aware… = I p 55

SUMMARY: SUMMARY: Undamped Free OSCILLATION x(t)=Real part of z(t); z(t) = x+iy A=Complex amplitude 56

Superposition of Two SHMs 57

Additions of two SHMs becomes convenient with imaginary exponents With, [] + Superposition of SHMs in 1D Case I- oscillation frequencies are the same 0 58

Case II- oscillation frequencies are different For simplicity we write the solutions as, Assuming that  2 >  1 Resulting displacement: Superposition of SHMs in 1D 59

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Superposition of two SHM in perpendicular directions Case I: 61

Square and sum the above two equations to obtain: General equation for an ellipse In most general case the axes of the ellipse are inclined to the x and y axes. 62 Eliminate ‘t’

Straight line Circle Ellipse otherwise The equation simplifies when the phase difference: 63

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Case II: Lissajous figures 65

SHO-266 Lissajous figures

Damped Free Oscillations 67

In many real systems, non-conservative forces are present. The system is no longer ideal. Friction/drag force are common non- conservative forces. In this case, the mechanical energy of the system diminishes in time, the motion is said to be damped. * A conservative force is a force with the property that the work done in moving a particle between two points is independent of the path taken. Damping (free oscillation case) 68

Damping (free oscillation case) Damping occurs due to coupling of energy of macroscopic oscillator to its surroundings, even if weakly Dissipation of energy takes place. We assume: viscous damping force or drag force that acts in opposite to the velocity 69

Examples of Damped Harmonic Oscillations a)The mass experiences a frictional force as it moves through the air. b)When the mass oscillates horizontally attached to a string, then there exists frictional forces between mass and surface. c)There are resistive force acting on the charge in LC circuit, due to wires and internal resistance of the devices. 70

Resistive force is proportional to velocity Damped Free OSCILLATION Damped Free OSCILLATION Or sometimes given in the form... Where, 71 Where,  =r/m and

General solution = Complimentary + Particular solution Linear differential equation of order n=2 Linear differential equation of order n=2 inhomogeneous inhomogeneous homogeneous homogeneous or For Complementary solution : 1.Take trial solution : x=e mt, m is constant 2.m1, m2,……….will be the roots. If all roots are real and distinct, then solution x=c 1 e m1t +c 2 e m2t +……………. 3. If some roots are repeated, say m1 repeated k times, then solution will be (c 1 + c 2 t+ …..c k t k-1 )e m1t 4. If some roots are complex, (if a+ib then a-ib are roots)solution will be e at (c 1 cos(bt) +c 2 sin(bt)) +……… For Particular solution : For Particular solution : Trial solution to be assumed depending on the form of ƒ(t) Solution

Hyperbolic functions Ref: Wikipedia Just as the points (cos t, sin t) form a circle with a unit radius, the points (cosh t, sinh t) form the right half of the equilateral hyperbola. 73

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Solution The equation is a second order linear homogeneous equation with constant coefficients. Solution can be found which has the form: x = Ce pt where C has the dimensions of x, and p has the dimensions of T -1. Trivial solution Solving the quadratic equations gives us the two roots: The general solution takes the form: 76

Case I: Overdamped (Heavy damping) The square root term is +ve: The damping resistance term dominates the stiffness term. Let: Then displacement is: Now, if: 77

Non-oscillatory behavior can be observed. But, the actual displacement will depend upon the boundary conditions 78