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Waves

Waves. W Richards The Weald School. Circular Motion. Circular Motion. 1) Is this car travelling at constant speed? 2) Is this car travelling at constant velocity?. Centripetal Acceleration. V a. Δ V. V b. If the velocity is changing then it must be accelerating.

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Waves

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  1. Waves W Richards The Weald School

  2. Circular Motion

  3. Circular Motion 1) Is this car travelling at constant speed? 2) Is this car travelling at constant velocity?

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

  5. Radians 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

  6. Radians Calculate the following angles in radians: 15cm 1) 2) 1.5cm 2cm 6cm 3) 4) 2.05cm 50.24m 8m 5mm

  7. Centripetal Acceleration v1 v θ v1 v2 v2 v r θ Consider a circle: Δv If we assume θ is very small then v1 = v2 = v Therefore θ = Δv/v Also θ = vΔt/r Therefore Δv/v = vΔt/r Δv/Δt = v2/r a = v2/r

  8. More Exciting Equations From the last slide a = v2/r but F=ma so centripetal force F = mv2/r F = mv2/r The “angular speed” is the “angular distance” divided by time, or ω = θ/t 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 ω = 2π/T “Frequency” is how often something happens every second, so T = 1/f. Therefore ω = 2πf ω = 2πf For a whole circle, v = 2πr/T. However, T = 2π/ω. Therefore v = rω v = rω a = rω2 Acceleration a = v2/r, therefore a = rω2 F = mrω2 Finally, this must mean that F = mrω2

  9. Example questions • A disc spins twice per second. Calculate its angular speed. • Estimate the angular speed of the Earth. • Scoon spins a conker around his head using a 50cm long string. The conker has a mass of 0.1kg and he spins it with a velocity of 2ms-1. Calculate the centripetal force. • Calculate the velocity of a satellite moving with an angular speed of 7x10-5 rads-1 and at an altitude of 700km above the Earth (radius 6370km). • 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? • If the combined mass of Tom and his car is 1000kg calculate the centripetal force if his turning circle has a radius of 3m.

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

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

  12. SHM Graphs Displacement Time Velocity Time Acceleration Time

  13. The Maths of SHM Displacement Time As we’ve already seen, SHM graphs are “sinusoidal” in shape: Therefore we can describe the motion mathematically as: x = x0cosωt a = -ω2x v = -x0ωsinωt a = -x0ω2cosωt

  14. The Maths of SHM a x 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 = -ω2x Important – remember ω = 2π/T

  15. SHM questions 5 2 a x • Calculate the gradient of this graph • Use it to work out the value of ω • Use this to work out the time period for the oscillations a • 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. • The maximum displacement of the pendulum is 3cm. Sketch a graph of a against x and indicate the maximum acceleration. x

  16. SHM Maximum Values xmax = x0 (obviously) vmax = -x0ω (or max speed = ωx0) x = x0cosωt Consider our three SHM equations: v = -x0ωsinωt a = -x0ω2cosωt Clearly, the maximum value that sinωt can take is 1, therefore: amax = -ω2x0

  17. SHM periods: Two examples  l g T = 2π For a pendulum the only thing that affects the period is the length of the string:

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

  19. More questions • Define simple harmonic motion. • A pendulum in a grandfather clock has a period of 1 second. How long is the pendulum? • Luke sets up a 200g 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 10s what is the spring constant? • Nick is envious of this and sets up another system with a spring constant of 0.1Nm-1. If the spring oscillates every 8 seconds how much mass did he use? • Simon sets up a pendulum and records the period as being 3 seconds. He then lengthens the pendulum by 1m and does the experiment again. What is the new period?

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

  21. SHM: Energy change Equilibrium position Energy GPE K.E. Time

  22. Waves revision Watch a “Mexican Wave”

  23. 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)

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

  25. The Wave Equation V  f 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

  26. Some example wave equation questions • A water wave has a frequency of 2Hz and a wavelength of 0.3m. How fast is it moving? • A water wave travels through a pond with a speed of 1m/s and a frequency of 5Hz. What is the wavelength of the waves? • The speed of sound is 330m/s (in air). When Dave hears this sound his ear vibrates 660 times a second. What was the wavelength of the sound? • Purple light has a wavelength of around 6x10-7m and a frequency of 5x1014Hz. What is the speed of purple light? 0.6m/s 0.2m 0.5m 3x108m/s

  27. Resonance Bridge video Glass video Resonance occurs when the frequency of a driving system matches the natural frequency of the system it is driving.

  28. Damping Amplitude of driven system Driver frequency High damping High damping

  29. Travelling Waves Energy flux = (in Wm-2) Power (in W) P 4πr2 φ = Area (in m2) An “inverse square law” 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:

  30. Example questions • Harry likes doing his homework. His work is 2m from a 100W light bulb. Calculate the energy flux arriving at his book. • If his book has a surface area of 0.1m2 calculate the total amount of energy on it per second (what assumption did you make?). • Matt doesn’t like the dark. He switches on a light and stands 3m away from it. If he is receiving a flux of 2.2W what was the power of the bulb? • Matt walks 3m further away. What affect does this have on the amount of flux on him?

  31. 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:

  32. Polarisation

  33. Refraction Revision

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

  35. Finding the Critical Angle… THE CRITICAL ANGLE 1) Ray gets refracted 2) Ray still gets refracted 4) Ray gets internally reflected 3) Ray still gets refracted (just!)

  36. 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

  37. Wave diagrams 1) Reflection 2) Refraction 3) Refraction 4) Diffraction

  38. Diffraction More diffraction if the size of the gap is similar to the wavelength More diffraction if wavelength is increased (or frequency decreased)

  39. Sound can also be diffracted… 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.

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

  41. Diffraction depends on frequency… A low frequency (long wavelength) wave will get diffracted more, so the house can receive it…

  42. Phase Difference 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”

  43. Phase Difference What is the phase difference between each of these waves?

  44. Coherence 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.

  45. Coherence These waves have the same frequency and the same constant phase difference, so they are “coherent”

  46. Superposition 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”:

  47. Superposition patterns Consider two point sources (e.g. two dippers or a barrier with two holes):

  48. Superposition of Sound Waves

  49. Path Difference 1st Max Min Max Min 1st Max Constructive interference Destructive interference 2nd Max

  50. Young’s Double Slit Experiment D λ s O x x D xs D λ s λ = = Screen A

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