my phy4 revision notes

PHY4 Revision Notes By: Abdul Gomaa

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By: Abdul Gomaa 25/05/09

Chapter Circular Motion Simple Harmonic Motion Waves Quantum Universe

Page 2 3 6 9 13


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PHY4 Revision Notes

1. Circular Motion

• Circular path at constant speed • Direction changes so velocity changes (not magnitude) • The body is accelerating – Centripetal acceleration • Newton’s First – Centripetal Force • Force acts towards centre of the circle

During circular motion, NO WORK is done, since:

• Distance moved in direction of force is zero. • Force and acceleration are towards the centre. • Velocity is at right angles to these.

Therefore, there is NO Change in Kinetic Energy.

Definitions:

Frequency (f): Number of oscillations/revolutions per second /Hz Period (T): Time taken for once complete oscillation/revolution

f = 1/T Angular Speed (ω): Rate of change of central angle per second. “rad s 1 ”

Equations:

Centripetal acceleration = v 2 /r = rω 2 Centripetal Force = ma = mv 2 /r = mrω 2 ω = 2π /T = 2πf v = rω

Circular motion under gravity

Centripetal force is the resultant force between the tension and the weight.

Max at C – String is pulling towards the centre, it must provide centripetal force and overcome the weight. Tension = Centripetal + mg

Min at A – weight provides centripetal force so tension is minimum. Tension = centripetal – weight. If slow enough, weight = centripetal therefore tension = zero.


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Conical Pendulum

2. Simple Harmonic Motion

This is motion in which the acceleration is directly proportional to the displacement from a fixed point and is directed towards that point.

• Regular repeated to and fro motion oscillating around an equilibrium position • + and – displacement • Amplitude = max displacement • Cycle = + then – displacement back to equilibrium • Period = time for one cycle • Period does not change when amplitude dies away • For fast oscillations use Frequency, not Period • Acceleration α displacement • Acceleration and displacement are in opposite directions • Acceleration α –displacement, a α –x • a = ω 2 x displacement and acceleration have the opposite sign • Acceleration (and resultant force) act towards equilibrium position • Displacement varies sinusoidally with time • Force α a α –x

Equations

F = kx = ma a = (k/m)x = ω 2 x ω 2 = k/m ω = √(k/m)

Mass on a spring, T = 2π√(m/k) Pendulum, T = 2π√(l/g)

Tcosθ = mg Tsinθ = mv 2 /r

Tanθ =v 2 /rg


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Simple Harmonic Motion Systems

In phase: When two systems oscillating exactly in step, but have different amplitudes.

In the waveform below the oscillators are 0.5 oscillations apart, they are exactly out of phase – in Antiphase. The phase angle difference between them is π radians

Common SHM Quantities

f = 1/T , 1 cycle 2π radians, Period of SHM is the same as period for circular motion = T Therefore T = 2π/ω and F = ω/2π, from the SHM equation a = ω2x, a = (2πF) 2 x.

X = Xocos(2πFt) = Xocos(ωt) amax = (2πf) 2 xo = ω 2 xo vmax = 2πFxo = ωxo

Energy in a SHM System

The energy of the system is taken to be constant made up of kinetic (max at x= 0, where velocity is a max and zero at x = A because velocity at A is zero) plus potential which has the opposite characteristics of the kinetic.


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Resonance and Damping

• All oscillating systems have a natural frequency of vibration. • If a system is FORCED to oscillate at this frequency resonance occurs. • The system oscillates with maximum amplitude (this can be useful or very destructive)

Types of Damping

Light damping – amplitude of oscillations decreases slowly over several oscillations CRITICAL damping – oscillating mass returns to rest in the shortest possible time without any oscillation Overdamping – system does not oscillate

• In real life situations oscillations usually die out due to friction. • This is called DAMPING • Amplitude of oscillation gradually decreases BUT period and frequency stay the same.

• Damping introduces friction into a system, energy is removed from the oscillating system, amplitude and max speed of oscillations deceases.


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3. Waves

Waves convey energy and information along the direction of travel.

Wave pulse: short duration disturbance Continuous wave: repeated disturbances Travelling or progressive waves: travel out from the source

Wave Equation, v = Fλ

The inverse square law

• Waves get weaker as they spread out from a source • As distance increases, power is spread over a larger area • Energy flux (Intensity) is the power per unit area (Wm 2 ), defined as “Energy the

wave carries perpendicularly through unit area each second” • Energy Flux, Φ = power/area = P/4πr 2

Polarisation

• Electromagnetic waves are transverse – the disturbance is perpendicular to direction of wave travel.

• They can be POLARISED. This is a property of electromagnetic radiation • The direction of oscillation and the direction of travel all lie in one PLANE only • This can be demonstrated with a slinky spring, with light and microwaves.


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Superposition of waves

Principle

• When two or more waves are present simultaneously in the same place, the resultant wave is the sum of the individual waves.

• When two waves of the same amplitude and frequency travelling toward each other meet at the centre of their paths the resultant crest has an amplitude equal to that twice of the original waves

• Displacement of the waves is vector quantities. The displacement at any point is the vector sum of each of the waves at that point.

• The amplitude of the resultant wave depends upon the phase difference between the two waves.

• When waves are in phase they reinforce each other – CONSTRUCTIVE INTERFERENCE

• When waves are completely out of phase they cancel – DESTRUCTIVE INTERFERENCE

• With the water waves this appears as calm area and choppy areas.

• In order to achieve a stable wave pattern the waves must be of the same frequency and any phase difference must be constant. This is known as COHERENCE.

• If amplitudes of two waves are very different amplitudes one wave will dominate

• If they are of similar amplitudes then cancellation is noticeable

Path Difference

• For double slit interference the central maxima occurs when the 2 waves are in phase (path difference = 0).

• The second maxima will occur when the two waves are out of phase by 2π (path difference = one wavelength)

• The next maxima will occur when the path difference is 2λ etc • In general:

Ø For constructive superposition, path difference = nλ Ø For destructive superposition, path difference = (n+1/2)λ


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• In order to observe superposition of light we need 2 coherent light sources. • Light from a filament lamp is incoherent as each part of the filament produces

light waves with different phase relationships. • By using a small slit and a filter we can produce double sources

superposition as the diffracted light is coherent with itself

Set up of Young’s Double Slit Experiment

λ = wavelength of the light d = separation of the two slits s = separation of the fringes x = distance between slits and screen

If x is much bigger than s and d then: λ / x = s / d, or λ / d = s / x, therefore λ = xs / d

d

x

s S1

S2


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4. Quantum

Photoelectric emission

This is when electrons are emitted from matter after the absorption of energy from electromagnetic radiation. Photoelectric emission occurs at the threshold frequency. Each material has its own threshold frequency. Energy of an emitted photon depends on frequency NOT intensity.

• High intensity wave below threshold frequency – no emission • Low intensity wave above threshold frequency – some emission

Work function,φ minimum energy needed to remove an electron.

E = hF h = plank’s constant = 6.63x10 34

In order to change from Joule to eV, divide by 1.6x10 19 . φ = hF0 (threshold frequency)

Energy of photon = Energy needed to remove an electron + KE of the emitted electron Energy of photon = φ + 0.5mv 2

• The easiest electrons to eject are those near the surface • Other electrons will need more then this – so have less KE • For a given frequency photoelectrons are emitted with a range of energies with a

clear max value

Measuring the energy of a photoelectron

We can measure the energy of a photoelectron by making it run up an electrical hill.

This experiment uses a photocell to investigate the photoelectric effect. Light of various frequencies is incident on the cell and photoelectrons are emitted and then form an electric current. A white light source is shone through various coloured filters to produce a series of different frequencies of light falling on the photocell for each frequency the potentiometer is adjusted until the stopping voltage is reached (no current).

The electrical hill is just high enough to stop even the fastest electrons arriving – KE lost by fastest electrons = elec PE gained by electrons going up hill


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Einstein’s Photoelectric equation = hf = φ + QV

Plotting a graph of Vs against frequency for the experiment gives a graph with a straight line slope of gradient h/Q (showing increasing frequency = increasing Vs therefore more kinetic energy produced).

Photocurrent

• When electrons are emitted a current flows. • If the frequency of radiation is constant and intensity is increased MORE

electrons are emitted per second but they have the SAME maximum energy • Increasing intensity increases the photocurrent

Light’s Wave Particle Duality

• Young’s double slits showed it to be a wave • Einstein showed it to consist of photons (matter particles) • Therefore, light is made up of particles that have wave properties

The EM Spectrum

• Electromagnetic radiation can be described in terms of a stream of massless photons

• Each travelling in a wavelike pattern and moving at the speed of light. • Each photon contains a certain amount of energy, and all electromagnetic

radiation consists of these photons, the only difference between the various types of electromagnetic radiation is the amount of energy found in the photons.

• Radio waves have photons with low energies, microwaves have a little more energy than radio waves, infrared has still more, then visible, ultraviolet, Xrays, and ... the most energetic of all ... gammarays

E = hc/λ

Wavelength (m) Frequency (Hz) Energy (J)

Radio > 1 x 101 < 3 x 109 < 2 x 1024

Microwave 1 x 103 1 x 101 3 x 109 3 x 1011 2 x 1024 2 x 1022

Infrared 7 x 107 1 x 103 3 x 1011 4 x 1014 2 x 1022 3 x 1019

Optical 4 x 107 7 x 107 4 x 1014 7.5 x 1014 3 x 1019 5 x 1019

UV 1 x 108 4 x 107 7.5 x 1014 3 x 1016 5 x 1019 2 x 1017

Xray 1 x 1011 1 x 108 3 x 1016 3 x 1019 2 x 1017 2 x 1014

Gammaray < 1 x 1011 > 3 x 1019 > 2 x 1014


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How are they made?

• Radio – oscillating currents in aerials • Microwaves – directly produced in waveguides • Infrared – hot bodies, LEDs • Visible – very hot bodies, LEDs • Ultraviolet – ultra hot bodies, sparks, discharge tubes • Xrays – quickly decelerating high speed electrons • Gamma rays – nuclear decay • Cosmic rays – from distant parts of the universe

Quantum Definitions

The ground state of an electron, the energy level it normally occupies, is the state of lowest energy for that electron. There is also a maximum energy that each electron can have and still be part of its atom. Beyond that energy, the electron is no longer bound to the nucleus of the atom and it is considered to be ionised.

Ionisation energy: The energy required to free an electron completely from an atom. First excited state: Electron has gained energy and is above its ground state Excitation energy: Energy required to raise electron above the ground state

Electrons do not stay in excited states for very long they soon return to their ground states, emitting a photon with the same energy as the one that was absorbed. When an atom is in an excited state, the electron can drop all the way to the ground state in one go, or stop on the way in an intermediate level.

Spectra

• Transitions among the various energy levels are unique for each element because the energy levels are uniquely determined by the protons and neutrons in the nucleus.

• When the electrons of a certain atom return to lower energy levels from excited states, the photons they emit have energies and frequencies that are characteristic of that kind of atom.

• Each element gives out a characteristic range of frequencies of emitted radiations called the atomic spectrum.

• This gives each element a unique fingerprint, making it possible to identify the elements present in a container of gas, or a star

• The change of energy is the difference in energy of the 2 states between which the electron moves, E = E2 – E1


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• As the frequency of emitted radiation is given by E = hf • Where f is frequency and h is Planck’s constant • hf = E2 – E1 • Thus for each atom which a characteristic set of energy levels, a characteristic set

of frequencies of radiation are emitted when the atom is excited • Energy is put in when electrons move from lower to higher energy levels – energy

is given out when electrons move from higher to lower energy levels

Energy Levels

• When the electron leaves the atom (ionisation) it can be said to have 0 PE • Between this and the ground state are a range of energies • In these energy levels electrons have less total energy than when they are free –

these energies are labelled with negative values • Energy level diagrams are used to describe the possible energy stated for an atom • We can observe characteristic spectra in the lab using spectral lamps and

diffraction gratings – the discrete lines correspond to the discrete energy levels in the atom

Atomic Models

1. Bohr – electrons orbit the nucleus in discrete orbits. Change in energy = change in orbit

2. Schrodinger – energy levels due to electrons behaving like stationary waves in the atom. These waves correspond to areas around the nucleus where the electron will probably be found

Diffraction of electrons

• The pattern of rings produced in the electron diffraction apparatus looks like this!

• There are places on the screen where many electrons hit and places where non hit.

• Electrons have wave properties!

• λ = h/ρ = λ = h/mv

• A grating can be used to diffract an electron beam

• For the nth order image produced, if N = number of slits per metre then:

• nλ =sinθ/N, Where θ is the angle to the central maxima

• The wave properties of particle theory holds for all particles – even big ones like us!

• Particles are separate particles and the wave associated with them (wave function) describes the PROBABLITY of finding the particle in a particular place.

• Wave properties are only significant for particles of atomic size or smaller.


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5. Universe

Star Spectra

• We have seen how hot gases emit different spectra and we can view them through a diffraction grating – characteristic frequencies are emitted as excited electrons move from higher to lower energy levels – EMMISION SPECTRA

• Electrons also absorb photons of energy and are promoted to higher energy levels

• Looking at spectra from the sun we see dark lines – as light passes through cooler gases in the outer layers of the sun some energy is absorbed – electrons are promoted to higher energy levels

• Only certain frequencies are absorbed as the absorbed energy must match the energy level – ABSORPTION SPECTRA

• The missing frequency is what gives us the information about the elements present

Red Shift

Electromagnetic radiation emitted by a moving object also exhibits the Doppler effect.

The radiation emitted by an object moving toward an observer is squeezed; its frequency appears to increase and is therefore said to be blue shifted. In contrast, the radiation emitted by an object moving away is stretched or red shifted. Blueshifts and redshifts exhibited by stars, galaxies and gas clouds also indicate their motions with respect to the observer.

We can measure the velocity of an object by analysing the apparent wavelengths of spectral lines from it. Objects in motion compress the light waves in front of them making them appear more blue (blue shift), the light waves behind are stretched out and appear more red (red shift).

Amount of shift of wavelength is proportional to the component of velocity along the line of sight.

Can calculate the velocity of motion along the line of sight:

∆λ /λ0 = v/c, Where c is the speed of light, λ0 is the wavelength of the light as seen at rest, and ∆λ is the measured change in wavelength. Absorption bands in the spectra of distant galaxies seem to be shifted towards the red end of the spectrum


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This suggests that the galaxies are moving away from us and the universe is expanding!

• The fact that all the galaxies are moving away from us leads us to the idea they must have been much closer together at some point – BIG BANG

Hubble’s Law

More distant galaxies are receding away from us faster than closer galaxies. The Hubble constant has a value of (2 ± 1) x 1018 s 1 . Astronomical distances are normally given in light years.

• Age of universe = distance travelled/speed • Age of universe = d/v = d/Hd = 1/H • Age of universe = 5x10 17 s ~10 10 years

• The Hubble constant changes as a function of time depending on the precise cosmological models as the expansion of the universe slows due to gravitational attraction of the matter within it.

• Most models give an age of the universe of order 1/H. • The current value of the Hubble constant is hotly debated, with two opposing

camps generally getting values near the high and low ends • The uncertainty in the measurement of cosmological distances also gives us the

rather uncertain value

Fate of the universe

The universe's fate is intimately connected to its shape which depends on a number Omega: ratio of the average mass density of the universe to the critical value required to just maintain equilibrium.

• All matter exhibits a gravitational force of attraction on all other matter • As the universe expands work is done against these gravitational forces


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• As galaxies attract each other through gravity it is thought that the rate of expansion is probably decreasing slowly

• What will happen will depend upon how much mass is in space. • For the BIG CRUNCH to occur density will above the CRITICAL DENSITY • ~ 1x10 26 kg m 3 – about 6 atoms per m 3

• Lower than that – OPEN UNIVERSE

Open Universe

• An open universe, corresponding to omega less than one, will expand forever – but at a slower and slower rate.

• Matter will spread thinner and thinner. Galaxies will exhaust their gas supply for forming new stars, and old stars will eventually burn out, leaving only dust and dead stars.

• The universe will become quite dark and, as the temperature of the universe will approach absolute zero.

• The universe will not end, exactly, just peter out in a BIG CHILL.

Close Universe

• The expansion of a closed universe, with an Omega greater than one, will slow down until it reaches a maximum size, when it begins its inward collapse.

• Like a video of the Big Bang and expansion run backward, the universe will become denser and hotter until it ends in an infinitely hot, infinitely dense Big Crunch, perhaps providing the seed for another Big Bang

If Omega equals 1 exactly, then cosmic expansion will coast to a halt infinitely far into the future. The universe will not end in a Big Crunch nor expand into an infinite Big Chill, but will remain at equilibrium.

• The mass of what we can see makes up of only a fraction of the total mass • 90% of mass of universe cannot be seen • Dark Matter – undetectable – evidence from gravitational effects

• Small failed stars dim, distant, size between star and planet • "Baryonic" matter made of protons, neutrons and electrons that fails to emit

radiation detectable on Earth

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