slides intro

University of Bristol – 2nd Year Classical Physics [email protected]

Classical Physics - Mechanics and Waves‣ Welcome!!‣ This part of the course deals with classical mechanics and waves!

‣ We will study selected topics in these areas!‣ Chosen to be ‘important’, and relevant to real-world problems!‣ Allow you to extend your range of knowledge to (interesting) new areas!‣ Will often point the way to more advanced methods and topics!

‣ To be pursued in 3rd / 4th year courses in some cases!

‣ Caveat emptor!!‣ Course asymptotically approaches being bug-free (see bead on spoke problem later)!‣ Please be tolerant of any cock-ups rough spots.!‣ I will be seeking your feedback regularly to improve the course!

‣ Who is the guy at the front?!‣ Prof. Dave Newbold, head of particle physics!‣ Office 4.57; Tel: x88770; email: [email protected]

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Course Content1. Oscillations, normal modes and Waves in 3D

2. Mechanics introduction!

3. Central forces and orbits!

4. Mechanics in non-inertial frames!

5. Many-particle systems!

6. Rigid body motion

7. Interference!

8. Diffraction (qualitative treatment)

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Learning methods‣ This course is ‘not easy’!‣ The challenge is to absorb many new concepts in a short time!‣ If you can do this, the maths should not particularly challenge you!‣ This means that you will need to focus on outside study!

‣ You should spend longer on private study / problems than in the lectures!

‣ Materials!‣ These slides – handed out and available on Blackboard!‣ Homework problems & answers / ‘past papers’ – in Problems Book!

‣ Teaching!‣ Lectures. Please come to them. Please ask questions!!!!

‣ Please complete prescribed reading before the lectures!

‣ Homework problems – graded and returned to you (solutions available later)!‣ Group work sessions – use to go through exam-style problems!‣ Problems classes – you will present your solutions to worked problems!‣ ‘Office hours’: (usually) the hour following each lecture

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NB: Exa

m form

at ch

anged

this yea

r!


Homework Problems‣ Four kinds of problems!‣ Homework problems for grading!

‣ Schedule and deadline are in Problems Book – late work will not be marked!

‣ Examples to present in problems classes!‣ You will be asked to work through these at the front – bit longer!

‣ ‘Open ended’ or practice problems!‣ Some revision questions!‣ More advanced problems point the way to ‘interesting’ topics for those who want to pursue them!

‣ Past papers!‣ Selected questions will be worked through at the small group sessions!‣ The rest you can complete for revision in your own time!

‣ Tackling the problems!‣ Homework problems: individual work!‣ Problems classes: work in groups (assignments will be posted on Blackboard)!

‣ Solution sheets!‣ Available at the problems classes

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Textbooks‣ Why buy / use textbooks?!‣ I have only ~18 hours to present this material; textbooks have 300 pages each.!‣ Helpful to see new concepts presented several different ways!

‣ Recommended texts:!‣ “An introduction to mechanics”, Kleppner & Kolenkow (‘K&K’), 2nd ed!

‣ Probably the best all-round introduction to 1st and 2nd year mechanics!

‣ “Classical mechanics”, Kibble & Berkshire (‘K&B’)!‣ The real deal – read alongside K&K!

‣ Also (at a more advanced level):!‣ “Analytical Mechanics”, Hand & Finch / “Classical mechanics”, Goldstein!

‣ “Introduction to Modern Optics”, Fowles, 2nd ed.!‣ Covers the course, cheap. However, if you plan to do ‘modern optics’ next year, consider:!

‣ “Optical Physics”, 3rd ed, Lipson, Lipson and Lipson.!‣ “Optics”, 4th ed., Hecht (my favourite – but not the ‘international edition’)!‣ “Vibrations and Waves”, French!‣ You did buy a copy of the “Feynman lectures”, right?

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Survival Guide‣ Some advice!‣ To survive, you need to engage and sustain that engagement!‣ This year, the lectures run for an extended period!‣ Reminder: you should spend 3–4 hours in private study for every lecture!‣ There is no time to ‘catch up’ before the summer exam!

‣ Come to the lectures!

‣ Buy and use the textbooks!

‣ Hand in the homework problems!

‣ Carry out the prescribed reading before lectures!

‣ Do the problems before the problems classes!‣ When advised, preferably before the next lecture!

‣ Ask questions, in lectures, in office hours, and at classes��12


Section 1 – Oscillations & Waves‣ More topics on oscillations & wave motion!‣ Follows directly from last year’s course – look at your notes!!

‣ To be covered!‣ Oscillations and waves revision!‣ Superposition of oscillations!‣ Coupled oscillators & normal modes!‣ The 3D wave equation!

‣ Goals!‣ Understand the general nature of harmonic motion!‣ Become familiar with the concept of normal modes of oscillation!‣ Understand origin of the wave equation!‣ Recognise and be able to apply the wave equation in 3D!

‣ See also!‣ French Ch. 2, 5; Pain Ch. 2, 4; Hecht Ch. 2, 7

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This Section Dedicated To…‣ Galileo Galilee (1564 - 1642)!‣ The original ‘Renaissance man’!‣ ‘The father of science’!

‣ Key achievements!‣ The first to describe physics in terms of

pure mathematics!‣ A key proponent of experimental science!‣ Disentangled many of the mechanical

concepts followed up by Newton et al!‣ And by us, in this course!

‣ But also…!‣ Was usually broke!‣ Worked on technology, as much as science!‣ Got into (very) big trouble with the Pope!‣ Did not drop anything off anything

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The Simple Harmonic Oscillator‣ 1D SHO properties – recall:!‣ Any system where the the ‘restoring force’ is

linearly proportional to the ‘displacement’:!‣ The system oscillates about an equilibrium

position with angular frequency:!‣ The energy of the system is given by: !‣ Forced SHO always oscillates at the

driving frequency, independent of natural frequency!‣ A forced, damped harmonic oscillator exhibits resonant behaviour!

‣ Describing the motion!‣ General solution to the motion is:!‣ Can also be written as: ! ! !‣ The variable ‘x’ can be any physical quantity!

‣ Displacement, electric field, angle, current, pressure, etc.!

‣ Phasor diagrams are useful to visualise addition of oscillations

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! =

rk

m

mx+ kx = 0

E = T + V =1

2mx

2 +1

2kx

2

x = Re⇣Ae

i(!t+�)⌘

x = A cos!t+B sin!T


Superposition of Oscillations‣ Systems are often subject to more than one excitation!‣ For a linear system, displacement is the combination of the corresponding SHMs!

‣ Some complex or heavily-driven systems are not linear in response!

‣ The total response under two excitations has the form:!

‣ Excitations of equal frequency!‣ The frequency of response is the same as the excitations!‣ Ex 1.1: Show that ! ! ! ! ! ! ! ! !

‣ Excitations of unequal frequency!‣ In general, a complicated response results!‣ If excitations close in frequency, obtain beats!

‣ A modulated response at the average frequency!

‣ Ex 1.2: Show that for two equal-amplitude excitations of close frequency:

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xr = x1 + x2 = a1 cos(!1t+ �1) + a2 cos(!2t+ �2)

a2r = a21 + a22 + 2a1a2 cos(�1 � �2) tan �r =

a1 sin �1 + a2 sin �2a1 cos �1 + a2 cos �2

xr = 2a cos

✓!1 + !2

2

t

◆cos

✓!1 � !2

2

t

◆


Coupled Pendula‣ Can introduce a coupling between two oscillators!‣ Energy can be shared between the two pendula!‣ General motion is more complicated than independent oscillation!

‣ Analysing the motion!‣ Ex 1.3: Write down the coupled equations of motion of each pendulum!

‣ Denote the displacements of the bobs by x, y!

‣ The motion of each pendulum depends on the positions of both!‣ Can untangle this by introducing two new ‘coordinates’:!‣ Ex 1.4: Show that the motion reduces to SHM in these two coordinates!

‣ The combined oscillation!‣ Each ‘mode’ acts like an independent oscillator, with its own frequency!

‣ Important: energy is transferred between the pends. over time, but NOT between the modes!

‣ The combined motion (for our tuned system) is subject to beating!‣ Frequency of each pend. is the average of pend. and spring natural freq, is modulated!‣ This works because we tuned the spring and pendulum constants to be close...

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X = x+ y;Y = x� y


Normal Modes‣ General way of analysing motion of a coupled system!‣ Can write down ‘normal coordinates’ for which eqns. of motion look like SHM!

‣ These are functions of the physical coordinates of the system!

‣ Each coordinate has its own ‘normal mode’, with a characteristic frequency!‣ All components of the system vibrate with the same frequency!‣ Can identify these frequencies by looking for resonance behaviour!

‣ The overall motion of each component is a superposition of normal modes!‣ The instantaneous energy of each component may vary with time!

‣ Energy of the system!‣ Each mode is independent – no energy is shared between normal modes!‣ Can ‘populate’ several normal modes with energy at t=0!

‣ Or we can carefully set up initial conditions to excite only some normal modes!

‣ For the coupled pendulums: !‣ In general, each excited mode contributes

independently to the total energy of the system

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E = EX + EY =m

2

hX2 + !2

0X2 + Y 2 + !2

sY2i


Degrees of Freedom‣ How many normal modes does a system have?!‣ Simple pendulum: one mode; Coupled pendulums: two modes!‣ Number of modes generally related to number of degrees of freedom of system!

‣ Examples!‣ Two masses connected by three springs in 1D!

‣ Equivalent to the coupled pendulums, N=2!

‣ Three masses connected by two springs in 1D!‣ This system has two normal modes, N=2. Why?!

‣ Three masses connected by four springs in 1D!‣ Three normal modes , N=3. What are they?!

‣ Example 1.5: What are the normal modes of vibration of the CO2 molecule?!

‣ ‘Real’ systems!‣ Can be approximated by systems of coupled oscillators with large N!‣ In quantum systems, phonon modes correspond to classical normal modes!

‣ Separate modes of excitation which can travel as waves without cross-coupling

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Another Example: Compound PendulaA2 (16 marks). A double pendulum (see diagram) consists of two identical point masses, suspended via light inflexible rods of length l.

!(a) (3 marks) Explain briefly what is meant by a ‘normal mode’ of a vibrating system, and sketch the normal modes of the double pendulum. !(b) (5 marks) Using the small angle approximation, and noting that the masses are subject to forces both due to gravity and the tension in the rods, show that the coupled equations of motion of the two masses can be written as: !!!!(c) (6 marks) The normal coordinates of the system have the general form: !!!Where k has a different value for each mode. By writing down the most general equation of motion of the system, and comparing the solution in part (b), find the normal mode frequencies. !(d) (2 marks) Describe an experimental technique that could be used to directly measure the normal mode frequencies.

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!!θ1 = −

gl2θ1 −θ2( )

!!θ1 + !!θ2 = −

glθ2

θ1 + kθ2


Large N Systems‣ A familiar system with large N!‣ Imagine a massless string under tension T

with N equal masses distance d apart!‣ Ex 1.6: Show that the transverse restoring

force on the nth mass is given by: !‣ The system comprises a set of N coupled oscillators!

‣ What are the normal modes?!‣ We expect to find N normal modes with N normal frequencies!‣ Ex 1.7: What are the modes of the system for N=2?!‣ Ex 1.8: Show that for large N, this solution

describes the behaviour of the system:!‣ Also applies to ‘masses on springs’, e.g. atoms in a crystal!

‣ Characteristics of systems with large N!‣ Minimum frequency (all-in-phase mode)!‣ Maximum frequency (alternating-phase mode)

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!!yn =

Tmd

yn−1 − 2yn + yn+1( )

nn −1 n +11 N

ynyn −1

yn +1

θ1θ2

an = C sinn✓j !2j = 2!2

0(1� cos ✓j)

✓j =j⇡

N + 1

!0 =

rT

md


The Wave Equation‣ What happens when ! ! ?!‣ We can attempt to describe the behaviour of a continuous system!‣ Replace with: !‣ Ex 1.9: Show that these replacements

lead to a new equation:!‣ Where ρ is the mass density, since!

‣ This is known as the wave equation!‣ Should be familiar from last year’s course!

‣ Will not deal again this year with dispersion, standing waves, impedance, reflection, etc!

‣ Applies to a very wide range of physical systems, not just strings!‣ In fact, any system which obeys the above relation for the ‘displacement’ y!‣ The ‘displacement’ can be transverse (strings, EM waves) or longitudinal (springs, sound)!

‣ Solution is any function !‣ This function represents a ‘wave pulse’ which propagates along the +ve or -ve x direction!‣ v is the phase velocity in the medium for a monochromatic wave!

‣ Note that the wave equation is linear!‣ Any superposition of wave pulses ! ! ! ! ! ! is a solution, and will travel as a wave

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N ! 1

yr ! y(x, t) yr ! @

2y(x, y)

@t

2yr±1 ! y(x± �x, t)

@

2y

@t

2=

T

⇢

@

2y

@x

2m ! 0

y = f(x± vt)

= y1 + y2 + · · ·+ yn


Waves in 3D‣ Harmonic waves!‣ Often use harmonic waves (waves of a single frequency component)!

‣ Easy, and we know that any disturbance is a superposition of harmonic waves (Fourier analysis)!

‣ General expression for disturbance at 1D position x due to harmonic wave:!‣ Where k is the wave number, related to the wavelength by! ! ! such that !

‣ Waves in 2D and 3D!‣ Can replace the wave number by a wave vector:!

‣ For plane wave, wave vector points in dirn. of propagation, has magnitude !

‣ Note that we are only concerned with the real part here – later (QM) will need full complex wavefunction!

‣ 3D wave equation:!‣ ! ! ! ! ! ! ! ! ! Laplacian operator is:!

‣ Please note - plane harmonic waves are an idealisation!‣ There is no such thing as a plane or harmonic wave (why not?)!‣ There is also no such thing as a spherical wave (why not?)

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(x, t) = Ae

i(kx�!t)

k =2⇡

�v =

!

k

(x, y) = Ae

ik·r�!t

|k| = (k2x

+ k2y

+ k2z

)1/2

r2 =1

v2@2

@t2 r2 ⌘ @

2

@x

2+

@

2

@y

2+

@

2

@z

2


More on 3D Waves‣ 3D wave propagation!‣ Just like in 1D, any wave pulse can propagate:!

‣ i.e. an arbitrary, non-harmonic, non-plane, wave function is still a solution of the wave equation!

‣ A useful idealisation in (2D, 3D) is the harmonic (plane, spherical) wave!‣ On the problem sheet, you will use a scary-looking

Laplacian in 3D spherical polar coords to show that a harmonic spherical wave is given by: !‣ Where - sign indicates movement out from the origin, + sign indicates convergence!

‣ Irradiance (power per unit area per unit time) is given by: !‣ We ‘automatically’ recover the inverse-square law for outward-propagating spherical waves !

‣ Polarisation!‣ Transverse 3D waves can have their displacement in arbitrary direction!

‣ Always perpendicular to the direction of motion!

‣ A wave for which the oscillation is confined to a particular direction is polarised!‣ Any (complete) poln. can be described as linear comb. of orthogonal states!

‣ This is a very general concept, also used to describe the motion of particles with a spin direction

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(x, y, z, t) = f(↵x+ �y + �z ± vt)

(r, t) =A

rcos (k(r ± vt))

I / !2a2(r)v


Summary‣ Oscillations!‣ Very general behaviour of a wide range of physical systems!‣ Superposition of close-frequency oscillations can lead to beating!

‣ Normal modes!‣ Systems of coupled oscillators can undergo complicated motions!‣ These are always separable into independent normal modes of oscillation!‣ The number of normal modes depends on the number of degrees of freedom!‣ For very large d.o.f. we obtain characteristic behaviour!

‣ Wave motion!‣ The limiting case of coupled oscillators, for a continuous medium!‣ Arbitrary waves can propagate in 1D, 2D and 3D media!‣ Often useful to build up arbitrary waves from plane / spherical, harmonic waves!‣ 3D spherical waves obey the inverse-square law!‣ 3D transverse waves can be polarised so that displacement is in a fixed direction

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