1072020 Waves W Richards The Weald School Circular

10/7/2020 Waves W Richards The Weald School

Circular Motion 10/7/2020

Circular Motion 1) Is this car travelling at constant speed? 2) Is this car travelling at constant velocity? 10/7/2020

Centripetal Acceleration 10/7/2020 If the velocity is changing then it must be accelerating. . . Va ΔV This change in velocity is towards the centre of the circle so the acceleration and is towards the centre if the circle – “Centripetal Acceleration” Vb

Radians 10/7/2020 To further understand circular motion we need to use a different system for measuring angles: Old method New method s r Angle = 300 Angle = s/r rad

Radians 10/7/2020 Calculate the following angles in radians: 1) 1. 5 cm 15 cm 2) 2 cm 6 cm 3) 2. 05 cm 5 mm 4) 50. 24 m 8 m

Centripetal Acceleration Consider a circle: v 1 θ v 2 r θ 10/7/2020 v 2 Δv If we assume θ is very small then v 1 = v 2 = v Therefore θ = Δv/v Also θ = vΔt/r Therefore Δv/v = vΔt/r Δv/Δt = v 2/r a = v 2/r

More Exciting Equations From the last slide a = v 2/r but F=ma so centripetal force F = mv 2/r The “angular speed” is the “angular distance” divided by time, or ω = θ/t The total time period T for one revolution must therefore be the time taken to complete 2π revolutions, or ω = 2π/T 10/7/2020 F = mv 2/r ω = θ/t ω = 2π/T “Frequency” is how often something happens every second, so T = 1/f. Therefore ω = 2πf For a whole circle, v = 2πr/T. However, T = 2π/ω. Therefore v = rω Acceleration a = v 2/r, therefore a = rω2 Finally, this must mean that F = mrω2

Example questions 10/7/2020 1) A disc spins twice per second. Calculate its angular speed. 2) Estimate the angular speed of the Earth. 3) Scoon spins a conker around his head using a 50 cm long string. The conker has a mass of 0. 1 kg and he spins it with a velocity of 2 ms-1. Calculate the centripetal force. 4) Calculate the velocity of a satellite moving with an angular speed of 7 x 10 -5 rads-1 and at an altitude of 700 km above the Earth (radius 6370 km). 5) Tom drives his car in circles. If he drives with an angular speed of 1 rads-1 how many times will he make a complete turn in 10 seconds? 6) If the combined mass of Tom and his car is 1000 kg calculate the centripetal force if his turning circle has a radius of 3 m.

Simple Harmonic Motion Definition: simple harmonic motion is when acceleration is proportional to displacement and is always directed towards equilibrium. 10/7/2020

Simple Harmonic Motion 10/7/2020 Consider a pendulum bob: Let’s draw a graph of displacement against time: Equilibrium position Displacement “Sinusoidal” Time

Displacement SHM Graphs 10/7/2020 Time Velocity Time Acceleration Time

The Maths of SHM 10/7/2020 Displacement As we’ve already seen, SHM graphs are “sinusoidal” in shape: Time Therefore we can describe the motion mathematically as: x = x 0 cosωt v = -x 0ωsinωt a = -x 0ω2 cosωt a = -ω2 x

The Maths of SHM 10/7/2020 Recall our definition of SHM: Definition: simple harmonic motion is when acceleration is proportional to displacement and is always directed towards equilibrium. This agrees entirely with the maths: a = -ω2 x Important – remember ω = 2π/T a x

SHM questions a 5 1) Calculate the gradient of this graph x 2 10/7/2020 2) Use it to work out the value of ω 3) Use this to work out the time period for the oscillations 4) Howard sets up a pendulum and lets it swing 10 times. He records a time of 20 seconds for the 10 oscillations. Calculate the period and the angular speed ω. 5) The maximum displacement of the pendulum is 3 cm. Sketch a graph of a against x and indicate the maximum acceleration. a x

SHM Maximum Values 10/7/2020 x = x 0 cosωt Consider our three SHM equations: v = -x 0ωsinωt a = -x 0ω2 cosωt Clearly, the maximum value that sinωt can take is 1, therefore: xmax = x 0 vmax = -x 0ω amax = -ω2 x 0 (obviously) (or max speed = ωx 0)

SHM periods: Two examples 10/7/2020 For a pendulum the only thing that affects the period is the length of the string: T = 2π l g

SHM periods: Two examples 10/7/2020 For a spring there are two things that affect the period – the mass and the spring constant: T = 2π m k Where k is defined as “the force needed to extend the spring by a given number of metres” (units Nm-1): F = -kΔx

More questions 10/7/2020 1) Define simple harmonic motion. 2) A pendulum in a grandfather clock has a period of 1 second. How long is the pendulum? 3) Luke sets up a 200 g mass on a spring and extends it beyond its equilibrium. He then releases it and enjoys watching it bounce up and down. If the period is 10 s what is the spring constant? 4) Nick is envious of this and sets up another system with a spring constant of 0. 1 Nm-1. If the spring oscillates every 8 seconds how much mass did he use? 5) Simon sets up a pendulum and records the period as being 3 seconds. He then lengthens the pendulum by 1 m and does the experiment again. What is the new period?

SHM recap questions 10/7/2020 1) Define SHM and state “the golden SHM equation” 2) A body is performing SHM and is temporarily at rest at time t=0. Sketch graphs of its displacement, velocity and acceleration. a 3) A body is performing SHM as shown on this graph. Calculate its angular speed and its time period T. 4) What is this body’s maximum speed? 5 10 x 5) A 1 kg mass is attached to a spring of spring constant 10 Nm -1. The mass is pulled down by 5 cm and released. It performs SHM. Calculate the time period of this motion. 6) Describe the energy changes in this system as it bounces up and down. 7) Calculate the length of a pendulum if it oscillates with a period of 5 s.

SHM: Energy change Equilibrium position 10/7/2020 Energy GPE K. E. Time

Waves revision Watch a “Mexican Wave” 10/7/2020

Some definitions… 1) Amplitude – this is “how high” the wave is: 2) Wavelength ( ) – this is the distance between two corresponding points on the wave and is measured in metres: 3) Frequency – this is how many waves pass by every second and is measured in Hertz (Hz) 10/7/2020

10/7/2020 Transverse waves are when the displacement is at right angles to the direction of the wave… Displacement Transverse vs. longitudinal waves Displacement Direction Longitudinal waves are when the displacement is parallel to the direction of the wave…

The Wave Equation 10/7/2020 The wave equation relates the speed of the wave to its frequency and wavelength: Wave speed (v) = frequency (f) x wavelength ( ) in m/s in Hz in m V f

Some example wave equation questions 10/7/2020 1) A water wave has a frequency of 2 Hz and a wavelength of 0. 3 m. How fast is it moving? 0. 6 m/s 2) A water wave travels through a pond with a speed of 1 m/s and a frequency of 5 Hz. What is the wavelength of the waves? 0. 2 m 3) The speed of sound is 330 m/s (in air). When Dave hears this sound his ear vibrates 660 times a second. What was the wavelength of the sound? 0. 5 m 4) Purple light has a wavelength of around 6 x 10 -7 m and a frequency of 5 x 1014 Hz. What is the speed of purple light? 3 x 108 m/s

Resonance Bridge video Glass video Resonance occurs when the frequency of a driving system matches the natural frequency of the system it is driving. 10/7/2020

Amplitude of driven system Damping 10/7/2020 High damping Driver frequency

Travelling Waves 10/7/2020 Definition: A travelling wave (or “progressive wave”) is one which travels out from the source that made it and transfers energy from one point to another. Energy dissipation Clearly, a wave will get weaker the further it travels. Assuming the wave comes from a point source and travels out equally in all directions we can say: Energy flux = Power (in W) (in Wm-2) Area (in m 2) φ= P 4πr 2 An “inverse square law”

Example questions 10/7/2020 1) Harry likes doing his homework. His work is 2 m from a 100 W light bulb. Calculate the energy flux arriving at his book. 2) If his book has a surface area of 0. 1 m 2 calculate the total amount of energy on it per second (what assumption did you make? ). 3) Matt doesn’t like the dark. He switches on a light and stands 3 m away from it. If he is receiving a flux of 2. 2 W what was the power of the bulb? 4) Matt walks 3 m further away. What affect does this have on the amount of flux on him?

Polarisation Consider a single wave of light: If you looked at it “end on” it might look like this: And lots of them might look like this: 10/7/2020

Polarisation 10/7/2020

Refraction Revision 10/7/2020

Refraction through a glass block: 10/7/2020 Wave slows down and bends towards the normal due to entering a more dense medium Wave slows down but is not bent, due to entering along the normal Wave speeds up and bends away from the normal due to entering a less dense medium

Finding the Critical Angle… 10/7/2020 1) Ray gets refracted 3) Ray still gets refracted (just!) THE CRITICAL ANGLE 2) Ray still gets refracted 4) Ray gets internally reflected

10/7/2020 Uses of Total Internal Reflection Optical fibres: An optical fibre is a long, thin, _______ rod made of glass or plastic. Light is _______ reflected from one end to the other, making it possible to send ____ chunks of information Optical fibres can be used for _____ by sending electrical signals through the cable. The main advantage of this is a reduced ______ loss. Words – communications, internally, large, transparent, signal

Wave diagrams 10/7/2020 1) Reflection 2) Refraction 3) Refraction 4) Diffraction

Diffraction 10/7/2020 More diffraction if the size of the gap is similar to the wavelength More diffraction if wavelength is increased (or frequency decreased)

Sound can also be diffracted… 10/7/2020 The explosion can’t be seen over the hill, but it can be heard. We know sound travels as waves because sound can be refracted, reflected (echo) and diffracted.

Diffraction depends on frequency… 10/7/2020 A high frequency (short wavelength) wave doesn’t get diffracted much – the house won’t be able to receive it…

Diffraction depends on frequency… 10/7/2020 A low frequency (long wavelength) wave will get diffracted more, so the house can receive it…

Phase Difference 10/7/2020 Phase difference means when waves have the same frequency but oscillate differently to each other. For example: These two waves have different amplitudes but the same frequency and hit their peaks at the same time – they are “in phase” These two waves start opposite to each other – they are “in antiphase” or “out of phase by π radians”

Phase Difference What is the phase difference between each of these waves? 10/7/2020

Coherence 10/7/2020 Two waves are said to be “coherent” if they have the same frequency and the same constant phase difference. For example: These waves have a different frequency, so phase is irrelevant.

Coherence 10/7/2020 These waves have the same frequency and the same constant phase difference, so they are “coherent”

Superposition 10/7/2020 Superposition is seen when two waves of the same type cross. It is defined as “the vector sum of the two displacements of each wave”:

Superposition patterns 10/7/2020 Consider two point sources (e. g. two dippers or a barrier with two holes):

Superposition of Sound Waves 10/7/2020

Path Difference Constructive interference Destructive interference 10/7/2020 1 st Max Min 1 st Max 2 nd Max

Young’s Double Slit Experiment 10/7/2020 D λ s O x λ s = x D λ = xs D A Screen

10/7/2020 Interference Patterns from 2 slits Intensity Distance

10/7/2020 Interference Patterns from 1 slit Intensity Distance

Stationary Waves 3 nodes 2 antinodes 5 nodes 4 antinodes 10/7/2020

10/7/2020 Quantum Physics W Richards The Weald School “Quantum” = “a small packet of energy”

Introduction 10/7/2020 Some basic principles: 1) The wavelength of blue light is around 400 nm (4 x 10 -7 m) 2) The wavelength of red light is around 650 nm (6. 5 x 10 -7 m) 3) Therefore blue light is higher frequency than red light 4) Light is treated as being a wave. Therefore the amount of energy a light wave contains should depend on its intensity or brightness.

Photoelectric Emission 10/7/2020 Consider a gold-leaf electroscope… Now charge the top: 5000 V +

Photoelectric Emission 10/7/2020 Let’s put a piece of zinc on top: Now shine some UV light onto it: lt U et ol vi ra - - - Ultra-violet light is causing the zinc to emit electrons – this is “Photoelectric Emission”.

Some definitions… 10/7/2020 For zinc, this effect is only seen when UV light is used, i. e. when the light has a frequency of 1 x 1015 Hz or higher. This is called the “Threshold Frequency” and is generally lower for more reactive metals. Max Planck (1858 -1947) proposed that electromagnetic radiation, like light, comes in small packets. The general name for these packets is “quanta”. In the specific case of electromagnetic radiation, a quanta is called a “photon” and its energy depends on its frequency, not how bright it is. The amount of energy needed to release an electron from a metal is called the “work function” and is given the symbol φ. Generally, work functions are lower for more reactive metals.

Photoelectron Energy 10/7/2020 …and some energy is given to the electron as kinetic energy. - Some energy is needed to release the electron (the work function φ)… Photon Energy = work function + kinetic energy of electron

Calculating Photon Energy 10/7/2020 I think that the energy of a photon is proportional to its frequency, so E=hf, where h = Planck’s Constant = 6. 63 x 10 -34 Js. On the previous slide we said that… Photon energy = work function + kinetic energy of electron hf = φ + 1/2 mv 2

10/7/2020 Measuring the Energy of a Photoelectron Illuminate the electrode: Electrons are “stopped” by this voltage A V +

The “Hill” analogy 10/7/2020 To help us understand this further, let’s say the electron is like a ball rolling up a hill… The amount of potential energy the electron gains is equal to the amount of kinetic energy it had at the start. Negative electrode Vs - In electric terms, the voltage the electron can work against depends on how much energy it had. Energy of electron = QVs = 1/2 mv 2 …where Vs is the “stopping voltage” (i. e. the height of the hill it can go up before coming back down again).

Photon Energy 10/7/2020 Combining the previous two slides, we get: Photon energy = work function + kinetic energy of electron hf = φ + QVs Let’s rearrange to give us a straight line graph: Vs = h f – φ Q Q Vs Gradient = h/Q Threshold frequency Photon frequency

Photocurrent vs Voltage 10/7/2020 If this voltage is large and negative no electrons will be able to move “up the hill”, so current is zero. A V However, if the voltage is positive electrons will be “helped up the hill”: Photocurrent Vs “Saturation” – all the electrons that get emitted are received. Voltage

10/7/2020 Photocurrents for different light 1) Different intensities: Photocurrent Bright Dim Voltage Vs is the same, because the electrons are emitted with the same energy (due to the same frequency of light)

10/7/2020 Photocurrents for different light 2) Different frequencies: Photocurrent Red Blue Voltage Vs is different as electrons emitted by blue light will have more energy (as blue light is higher frequency). Notice that the intensities are the same – the different saturations are because more electrons are emitted by red (but with less energy).

Photocurrents 10/7/2020 Recall the equation W=QV… This equation states that the work done on an electron (of charge 1. 6 x 10 -19 C) as it moves through a potential difference of 1 V is given by: W = QV = 1. 6 x 10 -19 x 1 = 1. 6 x 10 -19 J This is called “the electronvolt”, i. e. the work done on one electron as it moves through 1 volt. We can convert J into e. V by dividing by 1. 6 x 10 -19. Convert the following work functions into electronvolts: 1) Caesium – 3. 11 x 10 -19 J 2) Sodium – 3. 78 x 10 -19 J 3) Zinc – 5. 81 x 10 -19 J

Spectra 10/7/2020 Consider a ball in a hole: When the ball is here it has its lowest gravitational potential energy. 5 J We can give it potential energy by lifting it up: If it falls down again it will lose this gpe: 5 J 30 J 20 J

Spectra 10/7/2020 A similar thing happens to electrons. We can “excite” them and raise their energy level: 0 e. V -0. 85 e. V -1. 5 e. V -3. 4 e. V -13. 6 e. V An electron at this energy level would be “free” – it’s been “ionised”. These energy levels are negative because an electron here would have less energy than if its ionised. This is called “The ground state”

Spectra 10/7/2020 If we illuminate the atom we can excite the electron: Q. What wavelength of light would be needed to excite this electron to ionise it? 0 e. V -0. 85 e. V -1. 5 e. V -3. 4 e. V Light Energy change = 3. 4 e. V = 5. 44 x 10 -19 J. Using E=hc/λ wavelength = 3. 66 x 10 -7 m -13. 6 e. V (In other words, ultra violet light)

Example questions 1) State the ionisation energy of this atom in e. V. 2) Calculate this ionisation energy in joules. 3) Calculate the wavelength of light needed to ionise the atom. 0 e. V -0. 85 e. V -1. 5 e. V -3. 4 e. V 4) An electron falls from the -1. 5 e. V to the -3. 4 e. V level. What wavelength of light does it emit and what is the colour? 5) Light of frequency 1 x 1014 Hz is incident upon the atom. Will it be able to ionise the atom? -13. 6 e. V 10/7/2020

Different view of atoms The Bohr Atom 0 e. V Electrons are only allowed to have discrete energy values and these correspond to changes in orbit. The Schrodinger Atom Electrons behave like stationary waves. Only certain types of wave fit the atom, and these correspond to fixed energy states. 10/7/2020 + Amplitude

Spectra Continuous spectrum Absorption spectrum Emission spectrum 10/7/2020

Emission Spectra Hydrogen Helium Sodium 10/7/2020

10/7/2020 Evidence about the origins of the universe… In other words, let us revise Red Shift. Logical.

10/7/2020 Source of light “Spectra”

10/7/2020 If you pass the light through a gas something different is seen… helium Some wavelengths of light are absorbed by the gas – an “absorption spectrum”.

If the light source is moving away the absorption spectra look a little different… Before helium After 10/7/2020

The absorption lines have all been “shifted” towards the longer wavelength end (red end)… This is called red shift. The faster the light source moves the further its light will be “shifted” Before After A similar effect happens with sound – this is called “The Doppler Effect” Hear Doppler Effect 10/7/2020

Doppler Effect 10/7/2020

10/7/2020 Light from different stars and from the edge of the universe also shows this “red-shift”. This suggests that the universe is expanding. This is the BIG BANG theory

Star Spectra Basically, “Red shift” is an apparent shift in wavelengths of light towards the red (higher wavelength) end of the spectrum. It occurs when the light source is moving away from us. This effect is known as “the Doppler effect”. 10/7/2020 Before After The speed of this movement can be calculated: Δf f = Δλ λ = v c

Hubble’s Law 10/7/2020 Edwin Hubble, 1889 -1953 Recession velocity Astronomers have observed Red Shift in lots of galaxies and deduced the fact that more distant galaxies are moving faster than closer ones. I took this a step further: x x x xx Distance to galaxy

Recession velocity Hubble’s Law 10/7/2020 x x x xx Distance to galaxy Using this evidence I concluded two things: that the universe is expanding AND the recession velocity is proportional to the galaxy’s distance from us, therefore: V = Hd …where H = Hubble’s Constant (2± 1 x 10 -18 s-1)

The End of the Universe 10/7/2020 Basically, how the universe will end depends on its “energy-mass density”. Size of universe Stephen Hawking Now Open universe Critical density Closed universe Time