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Physics of the 20th and 21st centuries Lectures:

Relativity – special, general (week 1,2) Cosmology (week 3) Recap of some “classical physics” concepts (week 4) Quantum physics (week 4) Nuclear and particle physics (week 5,6) some condensed matter physics (week 6)

Lab experiments: some of the following: Standing waves -- resonance Earth’s magnetic field Geiger Müller counter, half life measurement operational amplifier mass of the K0 particle e/m of electron Franck-Hertz experiment Hall effect Planck’s constant from LED’s

Homework problems problem solving, modeling, simulations website http://www.physics.fsu.edu/courses/Summer14/YSP

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Introduction:recap of important ideas of “classical physics”

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Important tidbits of classical physics

TodayConserved quantities:

oEnergy, momentum

Waves on a stringThermodynamics

Later (as needed)Angular momentumElectromagnetismWave equationInterference and diffraction

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Conserved quantities

In “isolated system”, some quantities are “conserved” “isolated” = no external influence (e.g. force)

quantities important because they are conservedConservation is related to an “invariance” against

certain transformationsThree important conserved quantities:

Energy (linear) momentumAngular momentum

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Work and energy Newton’s 2nd law:

Force F acting on object of mass m acceleration a, a = F/m

Object’s velocity changes under influence of force

work is done when an object moves while force

is acting on it : W = F • d F = (net) force acting on object; d = displacement of object while force is

acting; F is really the component of the force in the

direction of motion (both force and displacement are vectors)

If force perpendicular to displacement no work done

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Potential and kinetic energylifting object:

work done against gravitational force; raised object can drop down and do work (e.g. pull a cart)

i.e. raising object (doing work on it), increased its potential to do work “gravitational potential energy”;

falling of raised object: object is accelerated -- loses potential energy --

gains energy of motion - “kinetic energy”; object can do work by virtue of its motion.

quantitatively:

W = F h, F = m g W = m g h let object drop: kinetic energy K = mv2/2

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Momentum

Newton’s 2nd law: p = “momentum” Relativistically,

For system of particles: where p = total momentum, F= net external force acting

on the system of particles If no external force momentum does not change;

“momentum is conserved” When speeds are not negligible wrt speed of light, it is

the “relativistic” momentum mv which is conserved

( )dv d dp

F ma m mvdt dt dt

p mv

2

1, ,

1

vp mv mc

c

,dp

Fdt

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WAVES

wave = disturbance that propagates “disturbance” e.g., displacement of medium element from

its equilibrium position; propagation can be in medium or in space (disturbance of a

“field”); mechanical waves:

when matter is disturbed, energy emanates from the disturbance, is propagated by interaction between neighboring particles; this propagation of energy is called wave motion;

a traveling mechanical wave is a self-sustaining disturbance of a medium that propagates from one region to another, carrying energy and momentum.

http://en.wikipedia.org/wiki/Wavehttp://www.physicsclassroom.com/class/waves/http://www.physicsclassroom.com/calcpad/waves/http://physics.info/waves/

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The great wave (off Kanagawa)

http://upload.wikimedia.org/wikipedia/commons/0/0d/Great_Wave_off_Kanagawa2.jpg

Katsushika Hokusai (1760 – 1849)

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Mechanical waves Examples of mechanical waves

o waves on strings,o surface waves on liquids,o sound waves in a gas (e.g. in air),o compression waves in solids and liquids;

it is the disturbance that advances, not the material medium

transverse wavedisplacements perpendicular to direction of propagation;

longitudinal wave: sustaining medium displaced parallel to direction of propagation (e.g. sound waves, some seismic waves, compression waves in a bell);

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periodic wave motion periodic wave motion:

particles oscillate back and forth, same cycle of displacement repeated again and again;

(we only discuss periodic waves)

terms describing waves: crest of the wave = position of maximum

displacement (“highest point of the wave”) wavelength = distance between

successive same-side crests

http://www.qrg.northwestern.edu/projects/vss/docs/communications/1-what-is-wavelength.html

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Quantities describing waves, cont’d

frequency f = number of same-side crests passing by a fixed point per second

period T = time for one complete wave oscillation: period = 1/frequency

unit of frequency: 1 Hertz = 1Hz = 1/second amplitude = amount of maximum displacement (height of

crest above undisturbed position) wave velocity v = velocity of propagation of wave crest wave velocity (speed of waves) depends on properties of the

carrying medium; in general: speed of mechanical waves in solids greater than

in liquids, and greater in liquids than in gases. relation between speed, wavelength and frequency:

v = f , i.e. speed = frequency times wavelength

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Energy in a wave intensity of a wave is a measure of how much power is

transported to a point by the wave; intensity = energy flow per unit time, per unit area =

power per unit area, (where area = area perpendicular to propagation direction)

energy flow carried by wave: is proportional to the square of the amplitude and the square of the frequency;

“inverse square law of wave intensity”: the intensity of a wave is inversely proportional to the square of the distance from the source of the wave

I = P/(4R2) (source = object emitting the wave)

(I = intensity, P = total power emitted by source, R = distance from source)

(strictly speaking, only for point-like or spherically symmetric sources, or if size of the source much smaller than R)

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Superposition of waves, interference Superposition principle:

two or more waves moving through the same region of space will superimpose and produce a well-defined combined effect; the resultant of two or more waves of the same kind overlapping is the algebraic sum of the individual contributions at each point, i.e. the (signed) displacements (elongations) add.

Huygens' principle every point on a wavefront can be considered

as a source, emitting a wave; the superposition of all these waves results in the observed wave.

consequences: interference, diffraction

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Interference interference:

superposition of two waves of same frequency can lead to reinforcement (constructive interference) or partial or complete cancellation (destructive interference;

constructive interference: two waves “in phase”, (i.e. crests of two waves coincide in time) reinforce each other, resultant amplitude bigger than that of individual waves;

destructive interference: two waves “completely out of phase” (i.e. out of phase by 1/2 period, so that crests of one wave coincide with troughs of the other) cancellation; complete cancellation (extinction) if both waves have same amplitude.

phase differences can be caused by: differences in path length; given a path length difference, the phase

difference depends on the wavelength; travel time difference due to difference in speed in different media; reflection;

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Interference, cont’d

examples: colors of thin films (oil on water, soap bubbles) dead spots in auditorium diffraction grating:

o many narrow parallel slits spaced closely together; o every slit forms source for wave; o differences in path length from different slits to some point

in space phase difference wavelength dependent interference pattern;

o can be used to measure wavelength; interferometers:

o Michelson - Morley (used to measure “ether wind”)o Fabry - Perot

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Summary – mechanical waves Travelling mechanical waves are due to the propagation of a

disturbance (displacement from equilibrium position) in a medium Medium must have some kind of “stiffness” (or “elasticity”) which

causes restoring force and the coupling between neighboring elements of the medium

Speed of wave propagation depends on strength of the coupling between neighbors and the inertia (resistance against being accelerated away from the equilibrium position)

Propagation speed, wavelength and frequency are related by v=f

Propagation speed v = (K/), where K = stiffness coefficient and is the density

Transverse waves: motion of oscillating elements direction of wave propagation

Longitudinal waves: motion of oscillating elements direction of wave propagation (e.g. sound waves in air)

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Summary mechanical waves – (2)

For a string, the stiffness coefficient is given by the string tension T and the density is the linear mass density (mass per unit length) of the string:

Waves can superimpose, which can lead to extinction or reinforcement (constructive or destructive interference)

Doppler effect: when source and receiver (observer) approach (move away from) each

other, the received (observed) frequency is higher (lower) than the emitted one

http://www2.hawaii.edu/~plam/ph170_summer/L15/15_Lecture_Lam.pdf http://www.physicsclassroom.com/class/waves/u10l1c.cfm http://en.wikipedia.org/wiki/Wave

Tv

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Reflection of a wave pulse

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Standing Waves

When two sets of waves of equal amplitude and wavelength pass through each other in opposite directions, it is possible to create an interference pattern that looks like a wave that is “standing still.”

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There is no vibration at a node.There is maximum vibration at an antinode. l is twice the distance between successive nodes or

successive antinodes.

l

Parameters of a Standing Wave

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standing waves on a string: reflection of wave at end of string, interference of

outgoing with reflected wave “standing wave” nodes: string fixed at ends

displacement at end must be = 0 “(displacement) nodes” at ends of string not all wavelengths possible;

length must be an integer multiple of half-wavelengths: L = n /2, n = 1,2,3,…

possible wavelengths are: n = 2L/n, n=1,2,3,…

possible frequencies: o fn = n v/(2L), n=1,2,3,…. (remember v=f, f=v/)o called “characteristic” or “natural” frequencies of the string;

o f1 = v/(2L) is the “fundamental frequency”; the others are called “harmonics” or “overtones”

o http://phet.colorado.edu/sims/wave-on-a-string/wave-on-a-string_en.htmlo http://hyperphysics.phy-astr.gsu.edu/hbase/waves/string.htmlo http://www.acs.psu.edu/drussell/demos.html

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Strings on instruments

When you pluck a stringed musical instrument, the string vibration is composed of several different standing waves.

The lowest frequency carried by the string is called “first harmonic”, also called “fundamental frequency”

standing wave pattern of fundamental frequency:

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Harmonics, overtones

Fundamental, 1st harmonic f1

L=/2, =2L2nd harmonic,

1st overtone, L=, f2 = 2f1

3rd harmonic, 2nd overtone, L=3/2, =2L/3

f3 = 3f1

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SOUND Sound waves propagate in any medium that can respond elastically and

thereby transmit vibrational energy. sound waves in gases and liquids are longitudinal (alternating compression

and rarefaction); in solids, both longitudinal and transversal; speed of sound is weakly dependent of frequency; speed of sound in air 340m/s at 20o C; increases with temperature;

1500m/s in water; three frequency ranges of sound waves:

below 20 Hz: infrasonic 20 Hz to 20 kHz: audible, i.e. sound proper above 20 kHz: ultrasonic, “ultrasound”

pitch is given by frequency e.g. “standard a” corresponds to 440 Hz intervals between tones given by ratio of frequencies (e.g. doubling of

frequency - one octave) male voice range 80 Hz to 240 Hz for speech, up to 700 Hz for song; female voice range 140 Hz to 500 Hz for speech, up to 1100 Hz for song.

http://phet.colorado.edu/en/simulations/category/physics/sound-and-waves

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Standing waves in pipes Gas –filled pipe

excite density oscillations in gas wave – reflection at end, superposition standing wave

open end: pressure node (pressure must be = outside pressure), displacement antinode

closed end: pressure antinode, displacement node (wall = fixed end)

One open, one closed end:o Fundamental: L=/4

f1 = v/(4L)

f3 = 3 f1 = 3v/(4L)

fn = nf1 = nv/(4L), n=odd=1,3,5,7,…..

oOnly odd harmonics presento http://cnx.org/content/m12589/latest/

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Standing waves in open pipes

One open, one closed end:

o Fundamental: L=/4 =4L

f1 = v/(4L)

f3 = 3 f1 = 3v/(4L)

o Only odd harmonics present

Both open:o Wavelengths,

harmonics just as for string with fixed ends

o Fundamental L = /2 =L/4

f1 = v/(2L)

f2 = 2f1 = 2v/(2L)

fn = 2f1 = nv/(2L)

http://www.electronicspoint.com/standing-waves-and-resonance-t222711.html http://hyperphysics.phy-astr.gsu.edu/hbase/waves/opecol.html http://www.physicsclassroom.com/class/sound/u11l5c.cfmhttp://www.s-cool.co.uk/a-level/physics/progressive-waves/revise-it/standing-waves-in-pipes

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Summary: standing waves Standing waves:

caused by interference of outgoing and reflected wave “boundary conditions” determine allowed wave patterns nodes (points of zero displacement) at fixed ends, antinodes (points of max.

displacement) on open ends Example: strings, pipes .. String with fixed ends:

o allowed wavelengths n = 2L/n (n=1,2,3,..)

o Allowed frequencies (natural frequencies of string): fn = (v/2L)n = f1n

o “fundamental frequency” f1 =v/2L

o 1st overtone = 2nd harmonic: f2 = 2 f1 = 2(v/2L)

o nth harmonic fn = nf1 = n(v/2L)

Pipe with one open end: only odd harmonicso allowed wavelengths n = 4L/n (n=1,3,5,..)

o Allowed frequencies (natural frequencies of string): fn = (v/4L)n = f1n, with n = 1,3,5,…

o “fundamental frequency” f1 =v/4L

o 1st overtone = 3rd harmonic: f3 = f1 = 3(v/4L)

o nth harmonic fn = nf1 = n(v/4L), with n odd

o http://phet.colorado.edu/sims/wave-on-a-string/wave-on-a-string_en.htmlo http://hyperphysics.phy-astr.gsu.edu/hbase/waves/string.htmlo http://www.acs.psu.edu/drussell/demos.html

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Experiment 1 experiment to determine the speed of sound in air.

put a loudspeaker above a large empty graduated cylinder and try to create resonance. The air column in the graduated cylinder can be adjusted by putting water in it. For each frequency, find a water column height for which a clear resonance is heard.

adjust the water height finely to get the peak resonance, then carefully measure the air column from water to top of air column.

What to do:(1) Fill out the columns in the table (assume that the loudspeaker creates an anti-node and the

water creates a node.) (note that the values given in the table are for illustrative purposes – actual values will be determined in class and included in instructions on Blackboard)

(2) Determine uncertainty of your result for the speed of sound (std. deviation)

(3) Explain the inherent errors in this experiment

frequency (Hz) length of air column(cm) wavelength (m) speed of sound (m/s)

184 46    

328 26    

384 22    

440 19

512 16    

1024 9    

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Conservation of mechanical energy conservation of (mechanical) energy:

when lifting the object, its gravitational potential energy is increased by the amount of work done lifting;

Work done against gravitational force Fg when lifting object by height h: W = Fg h = mgh

when the object falls, this energy is converted (transformed) into “kinetic energy” (energy of motion) (provided there is no “loss” due to friction,..)

gravitational potential energy: Ug = m g ho m = mass, h = height to which object was lifted, g =

gravitational acceleration

kinetic energy K = ½ mv2

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Conservation of EnergyEnergy conservation:

the total energy of all participants in any process is unchanged throughout that process. Energy can be transformed (changed from one energy form to another), and transferred (moved from one place to another), but cannot be created or destroyed. In an isolated system the total amount of energy is conserved – i.e. neither decreases nor increases

Verification : need to keep track of energy in all of its forms

Whenever experiments seem to indicate loss or gain of energy, a new form of energy was postulated and found

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Types of energy – (1) Many different kinds of energy; can be transformed

back and forth into each other: kinetic energy = energy of motion

o Translational: motion of of an object of mass m and speed v , only correct if v<<c

o Rotational : associated with rotation of object around an axis, depends on angular velocity (radians/second) and moment of inertia I (depends on how mass is distributed wrt rotation axis)

o Moment of inertia :o Sum integral for continuous mass distributiono Examples:

Thin ring (all mass concentrated at distance R)

Solid sphere

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2transK mv

21

2rotK I

2i iI m r

2I MR

22

5I MR

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Types of energy – (2)

potential energy = energy of position or state; (gravitational, elastic, electric, chemical, nuclear)

Gravitational potential energy = energy due to gravitational force between massive particles;

o Near surface of Earth:

m = mass, g = grav. acceleration, = unit vector in upward direction

o In general: force between two massive objects:

potential energy

ˆgravF mgz

gravU mgz

1 22

ˆgrav

GmmF r

r

1 2grav

GmmU

r

z

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Types of energy – (3)

Electrostatic potential energy: energy due to electric force between charged particles

o Coulomb force between two charged objects:

potential energy

elastic energy due to ability of deformed (stretched, squeezed,..) system to snatch back (e.g. rubber band, spring..)

o spring force o Potential energy

Force vs potential energy:o Change in potential energy = -(work done by the force);o Only change in potential energy is determined, U can

be set =0 at convenient position

1 2. . 2

ˆpt ch

kq qF r

r

1 2.pt ch

kq qU

r

F x

0

1

4k

21

2elasticU x2

2 1

1

, F U U U Fd

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Types of energy – (4) chemical energy = energy stored in molecular

structure of chemical compounds; can be “liberated” by chemical reactions converting compound into other compounds with less stored chemical energy.

thermal energy = kinetic energy of random motion of molecules; brought into system by “heating”; different from other forms of energy - not all of it can be converted back.

electromagnetic energy (electric energy) = energy due to electromagnetic forces;

radiant energy = energy carried by electromagnetic radiation;

nuclear energy = energy due to nuclear structure, i.e. how protons and neutrons are bound to each other to form nuclei

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Internal energy

For most classical purposes, internal energy = sum of microscopic kinetic and potential energies of all atoms and molecules of an object.

Object’s overall kinetic and potential energies can be converted into increased random internal energy (e.g. dropping book to the floor, friction, non-perfectly elastic deformation,..)

100% conversion overall internal is possible, but 100% conversion internal is not (2nd law of thermodynamics)

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Conservation laws

Conservation laws in physics “conserved quantities”: = quantities that do not change - “are

conserved” Conservation laws are related to “symmetry” property of system -

also called “invariance” property. Every invariance property is associated with a conserved quantity. Energy conservation is related to “invariance under translation in

time” (i.e. laws of physics do not change as time passes).

Other conserved quantities: omomentum (invariance under translation in space); oangular momentum (rotation); oelectric charge (“gauge transformation”); ocertain properties of subatomic particles (e.g.

“Isospin”, “color charge”, ...)

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Momentum

Newton’s 2nd law: p = “momentum” Relativistically,

For system of particles: where p = total momentum, F= net external force acting

on the system of particles If no external force momentum does not change;

“momentum is conserved” When speeds are not negligible wrt speed of light, it is

the “relativistic” momentum mv which is conserved

( )dv d dp

F ma m mvdt dt dt

p mv

2

1, ,

1

vp mv mc

c

,dp

Fdt

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Rotation, angular velocity, torque

For extended rigid object, can have rotation in addition to translational motion

acceleration of center of mass of object

Analogon to Newton’s 2nd law: I = moment of inertia, = angular acceleration, =

angular velocity Torque r = “moment arm” = vector from rotation axis to point of

application of force Direction of given by right-hand rule (fingers curl in

direction of rotation, thumb points in direction of ) http://hyperphysics.phy-astr.gsu.edu/hbase/rotv.html

, total cm cmexternalF m a a

, , cm cm cmexternal

dI

dt

r F

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Rotation, torque, angular momentum

http://hyperphysics.phy-astr.gsu.edu/hbase/rotv.html#rvec2

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Angular momentum

Rotational 2nd law:L = angular momentum, in many cases

analogous to Torque cause change or angular momentumIf no net torque, L does not change --- angular

momentum is conservedIf mass distribution changes moment of inertia

changes must changeExample: point mass at distance

r from rotation axis:

, , dL

L r pdt

, L I

, p mv

2tang , | | sin sin ( )L r p L rp rmv rmv rm r mr

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Thermodynamics

Describe systems of many particles whose microscopic behaviors are out of our control

Energy:Thermodynamic system can exchange energy

with its surroundings by:oVia workoVia heat input/output

1st law of thermodynamics:Einternal = Q – W

oQ = heat added to the systemoW = work done by the system

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Entropy “Disordered” arrangement of particles is more probable

than ordered one Entropy = measure of the “multiplicity” of a

thermodynamical state, i.e. of the number of microstates that give rise to the same “macrostate” (i.e. same number of particles, same temperature,…) lnBS k W

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Entropy, Temperature Boltzmann:

kB = Boltzmann’s constant

W= number of microscopic ways in which the particle distribution can be obtained (related to probability of a state)

2nd law of thermodynamics:o For an isolated system, entropy does not decrease

Temperature of a thermodynamic system in equilibrium defined by Adding random energy to a system (heat) raises its

disorder/entropy, temperature = dQ/dS

Entropy is function of a system’s internal energy E, volume V, number of particles N; strictly speaking

dQdS

T

0S

1S

TE

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Equipartition theorem, average energy

Amount of energy in certain internal forms: kinetic (translational), potential, rotational,..

Every way of “storing” internal energy = “degree of freedom”

Equipartition theorem: for every degree of freedom, the average internal energy per particle = ½ kB T

e.g. average translational kinetic energy in 3 dimensions:3 indep. variables 3 degrees of freedom Ktransl

= 3/2 kB T

2 2 2 21 1( )

2 2 x y zK mv m v v v