Oscillations and Waves

1. Oscillations and Waves

3. What is a wave?

4. How do the particles move?

5. Some definitions… 1) Amplitude – this is “how high” the wave is: Define the terms displacement, amplitude, frequency and period. 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)

6. Describing waves 1 cycle is described by 2π radians of phase Displacement T Hyperlink T Time 2π ω = 2π/T f = 1/T ω = Angular frequency f = Frequency T = Time period ω = 2πf Hyperlink and scroll down What are radians?

7. Phase and angle

8. Examples of phase difference

9. Oscillations http://www.acoustics.salford.ac.uk/feschools/waves/shm.htm#motion

11. Simple harmonic motion Any motion that repeats itself after a certain period is known as a periodic motion, and since such a motion can be represented in terms of sines and cosines it is called a harmonic motion. The following are examples of simple harmonic motion: a test-tube bobbing up and down in water (Figure 1)a simple penduluma compound penduluma vibrating springatoms vibrating in a crystal latticea vibrating cantilevera trolley fixed between two springsa marble on a concave surfacea torsional pendulumliquid oscillating in a U-tubea small magnet suspended over a horseshoe magnetan inertia balance

12. Data loggers • Use the data loggers to find the variation of displacement with time for an oscillating mass on a spring. • Process this data to find velocity and acceleration with time. • Now use the data to obtain a graph of Acceleration versus displacement.

13. Analysing your graphs • From the graph, find the • Time period • Angular frequency • Amplitude • The value of the velocity at maximum displacement • The value of the acceleration at zero displacement • The relationship between displacement and acceleration.

14. SHM definition • Find the gradient of your graph of acceleration and displacement. • Use this to calculate ω • Calculate T from the graph. • How the graph fit in with the definition of SHM?

15. SHM

16. Free body diagram for SHM • Draw the free body diagram for the mass when it is • in the centre of the motion • At the top of the motion • Between the bottom and the middle, down • Between the bottom and the middle, heading upwards. Hyperlink http://www.acoustics.salford.ac.uk/feschools/waves/shm2.htm

17. Spring pendulum Spring Pendulum Hyperlink

18. Oscillations to wave motion

19. Restoring forces

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

21. Pendulum Simple Pendulum Hyperlink

22. SHM Graphs Displacement Time T Velocity Time Acceleration Time

23. Definition of SHM Acceleration Displacement Now write your OWN definition of SHM

24. Displacement Time The Maths of SHM Therefore we can describe the motion mathematically as: x = x0cosωt a = -ω2x v = -x0ωsinωt a = -x0ω2cosωt

25. Students are expected to understand the significance of the negative sign in the equation and to recall the connection between ω and T. ω = 2π/T

26. a x SHM questions 5 • 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 2 a • Ewan 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

27. Questions • Q’s 1 – 9 from the worksheet • Using the equations • V = V0cosωt • V = V0sinωt • x = x0cosωt • x = x0sinωt • V = ± ω√(x02-x2) When do you use cos or sin?

29. 4.2 Energy changes during simple harmonic motion (SHM) At which points are -max displacement? -max velocity? -max acceleration? - max Ek -max Ep -max total energy? Total energy Energy -x0 x0 Displacement (x)

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

31. Energy formulae Ek = ½ mω2(x02 – x2) Ep = ½ mω2x2 Etotal = ½ mω2x02 Total energy Energy -x0 x0 Displacement (x)

32. Questions

33. Answers

34. 4.3 Forced oscillations and resonance 4.3.1 State what is meant by damping. “It is sufficient for students to know that damping involves a force that is always in the opposite direction to the direction of motion of the oscillating particle and that the force is a dissipative force.”

36. Free and Forced oscillations

37. Forcing frequency too slow

38. Forcing frequency too fast

39. Forcing frequency equals natural frequency

40. Resonance

41. Resonance and frequency Hyperlink

42. Resonance and frequency The width of the curve (Q value) is determined by the damping in the system. The value of the resonant frequency depends factors such as the size of the object…..

43. Tacoma Narrows

44. Useful resonance • Musical instruments • Microwave ovens • Electrical resonance when tuning a radio

45. Damping

46. Damped oscillations

47. Amplitude of driven system Driver frequency Damping Low damping High damping

48. Damping How much damping is best?

49. Critical damping

50. Wave characteristics The wave pulse transfers energy Students should be able to distinguish between oscillations and wave motion, and appreciate that in many examples, the oscillations of the particles are simple harmonic. If the source continues to oscillate, then a continuous progressive wave is produced.