PH4

1. PH4 Vibrations • Simple harmonic motion - s.h.m.

3. a mass bouncing on a spring • Examples include….

4. …or bungee jumping

5. …and a swinging pendulum

6. Simple harmonic motion is a special type of repetitive motion….. • The time period of the oscillation stays the same even if the amplitude varies. • The time taken to get from a to b and back to a in all three cases below is the same.

7. Also… • … the acceleration of the body is directly proportional to its displacement from a fixed point • and is always directed towards that point.

8. Let’s consider a pendulum…(taking positive to be to the right) • Displacement, x = max = amplitude, A • Acceleration, a = max = -amax (left) • Velocity = zero • So… • x = xmax = A • a = -amax • v = 0

9. As it swings through the centre… • Displacement, x = 0 • Acceleration, a = 0 • Velocity = max = -vmax (left) • So… • x = 0 • a = 0 • v = -vmax

10. It stops and then… • Displacement, x = max = amplitude, -A • Acceleration, a = max = amax (right) • Velocity = zero • So… • x = xmax = -A • a = amax • v = 0

11. As it swings through the centre again… • Displacement, x = 0 • Acceleration, a = 0 • Velocity = max = vmax (right) • So… • x = 0 • a = 0 • v = vmax

12. So the acceleration is always doing what the displacement is doing…they are directly proportional • x = xmax • a = amax • v = 0 • x = xmax • a = -amax • v = 0 • x = 0 • a = 0 • v = vmax • x = 0 • a = 0 • v = -vmax

13. Similarly with a mass on a spring…

14. So the defining equation for shm is… • The minus sign means the acceleration and displacement are oppositely directed.

15. ..and the definition in words is… • If the acceleration of a body is directly proportional to its distance from a fixed point and is always directed towards that point, the motion is simple harmonic.

16. Simple harmonic motion can be characterized by a sine function.

17. The bigger the amplitude of the oscillation the higher the peak of the sine wave

18. The usual equations and terms apply T • T (s) = Time Period = time for one oscillation • f (hz) = frequency = no. of oscillations/sec • A = amplitude = maximum displacement from equilibrium position A t

19. Plotting the pendulum’s displacement displacement time

20. Consider the pendulum again… • Starting with the pendulum pulled up to the right… • Displacement is a maximum (equal to the amplitude) …and velocity is zero…

21. Then displacement decreases as… • …velocity increases, but to the left, so in the negative direction.

22. Putting all three together… • Displacement and velocity we’ve talked about… • …and acceleration and displacement do the same as each other but in opposite directions i.e. both go from max to min but in opposite directions.

23. Timing your pendulum…(as you do) • You can start timing from when x = A • Or from when x = 0 • These give different displacement-time graphs t=0 x=0 t=0 x=-A

24. Cosine curve t=0 x=A x = Acos(t + ) t=0 x=0 Sine curve x = Asin(t + )

25. All the equations!

26. Period, T (s) = time for one oscillation • Frequency, f (Hz) = number of oscillations per second • Angular frequency,  (rad/s) = 2f

28. The auxiliary circle • Relating circular motion to simple harmonic motion or wave motion

30. Maximum values VERY important!

31. Energy -A +A Is she SERIOUS? The Fish diagram! Max k.e. Max p.e. k.e. = 0 Fish eye!! p.e. = 0

32. a x Graph of acceleration against displacement Think …how are these related? They’re directly proportional to each other … but oppositely directed +2A +A -A -2A

33. Phase constant,  • x = Acos(t + ) is the general solution to the equation d2x/dt2 = - 2x • where  is a constant phase whose value is determined by the position of the oscillator at t = 0. • For example, if x = 0 at t = 0,  = -/2 and x = Acos(t - /2) = Asin t

34. Hooke’s Law F = -kx • Force is proportional to extension. • Minus sign because the restoring force and the extension are oppositely directed. • K, spring constant, a measure of the stiffness of the spring = F/x i.e. the force required to produce unit extension.

35. Bouncing spring - is it shm? • Suspend a mass m on a spring, pull it down a distance x below equilibrium and release. • The weight of the mass is supported by tension in the spring when it is in equilibrium i.e. W = mg • If it is displaces a distance x below equilibrium the spring tension increases by an amount kx • There is a restoring force kx on the mass F = -kx

36. Mass on a spring • The motion is simple harmonic because f  -x (f will cause a) • a = -2x F = ma so F = -m 2x • Since F = –kx k = m 2 • T = 2/ 2 = k/m so

40. Damping • When a bell rings it is transferring energy stored in its oscillation to sound by moving the air around. • It does work against frictional forces and is said to be damped. • Whenever frictional forces act on an oscillator its total energy will diminish with time so that its amplitude decays to zero.

41. Damping • The heavier the damping (larger frictional forces) the greater the rate of decay.

42. Damped oscillations • So damped oscillations are when the amplitude of the oscillations becomes gradually smaller and smaller as energy is taken out of the system.

43. Deliberate damping! • Damping is deliberately introduced into some systems to prevent continuous oscillations. • An example is car shock-absorbers.

44. Free and Forced oscillations • Free oscillations – the system oscillates without any force applied. • Its frequency is its natural frequency and there is little or no damping. • Forced oscillations – the system responds to a regular periodic driving force – like continually pushing a child on a swing.

45. Resonance • If the frequency of the applied force equals the natural frequency of the system resonance occurs. • This is when the system oscillates with MAXIMUM amplitude. • If you push the child on the swing each time they reach maximum amplitude their oscillation amplitude increases . If you push when they’re half way back towards you their oscillation amplitude decreases.

46. Resonance • A good example of resonance is when a singer sings a note of frequency equal to that of the natural frequency of a wine glass….

49. Resonance and engineering.

50. Critical damping • The system, when displaced and released, returns to equilibrium, without overshooting as quickly as possible. • Useful in car shock absorbers – cars mustn’t go into resonant oscillation when they go over bumps in the road. So shock absorbers critically damp the oscillations once they have started.