1. Chapter 12: Vibration and Waves 12.1 Simple Harmonic Motion

2. Simple harmonic motion Periodic motion: Back and forth motion over the same path –E.g. Mass attached to a spring k m

3. Simple Harmonic Motion

4. Simple harmonic motion At the unstretched position, the spring is at equilibrium (x=0) The spring force increases as the spring is stretched away from equilibrium As the mass moves towards equilibrium, force (and acceleration) decreases

5. Simple harmonic motion Momentum causes mass to overshoot equilibrium Elastic force increases (in the opposite direction)

6. Simple harmonic motion Defined as a vibration about an equilibrium position in which a restoring force is proportional to the displacement from equilibrium The force that pushes or pulls the mass back to its original equilibrium position is called the restoring force Spring force = - (spring constant x displacement) Hooke’s Law:

7. Hookes Law Example Example 1: If a mass of 0.55kg attached to a vertical spring stretches the spring 2 cm from its equilibrium position, what is the spring constant? Given: m = 0.55 kg x = -0.02 m g = -9.8 m/s 2 Solution: F net = 0 = F elastic + F g 0 =- kx + mg or, kx = mg k = mg/x = (0.55 g)(-9.8 m/s 2 )/(-0.02 m) = 270 N/m x = -0.02 m FgFg F el

8. Energy What kind of energy does a springs has when it is stretched or compressed? –Elastic Potential energy Elastic Potential energy can be converted into other forms of energy –i.e. Bow and Arrow

9. The Simple Pendulum Consists of a mass, which is called a bob, which is attached to a fixed string Assumptions: –Mass of the string is negligible –Disregard friction

10. The Simple Pendulum The restoring force is proportional to the displacement The restoring force is equal to the x component of the bob’s weight When the angle of displacement is >15 o, a pendulums motion is simple harmonic

11. The Simple Pendulum In the absence of friction, Mechanical energy is conserved

12. Simple Harmonic motion

13. Chapter 12: Vibration and Waves 12.2 Measuring simple harmonic motion

14. Amplitude, Period and Frequency Amplitude: The maximum displacement from the equilibrium position Period (T): The time it takes to execute a complete cycle of motion Frequency (f): the number of cycles/vibrations per unit time

15. Period and Frequency If the time it takes to complete one cycle is 20 seconds: –The Period is said to be 20s –The frequency is 1/20 cycles/s or 0.05 cycles/s –SI unit for frequency is s -1 a.k.a hertz (Hz)

16. Measures of simple harmonic motion

17. The period of a simple pendulum Changing mass does not change the period –Has larger restoring force, but needs larger force to get the same acceleration Changing the amplitude also does not change the period (for small amplitudes) –Restoring force increases, acceleration is greater, but distance also increases

18. The Period of a simple pendulum LENGTH of a pendulum does affect its period –Shorter pendulums have a smaller arc to travel through, while acceleration is the same Free-fall acceleration also affects the period of a pendulum

19. The Period of a mass-spring system Restoring force –Not affected by mass Increasing mass increases inertia, but not restoring force --> smaller acceleration

20. The Period of a mass-spring system A heavier mass will take more time to complete a cycle --> Period increases The greater the spring constant, the greater the force, the greater the acceleration, which causes a decrease in period

21. Chapter 12 12.3 Properties of Waves

22. Wave Motion Lets say we drop a pebble into water –Waves travel away from disturbance –If there is an object floating in the water, it will move up and down, back and forth about its original position –Indicates that the water particles move up and down

23. Wave Motion Water is the medium –Material through which the disturbance travels Mechanical wave –A wave that propagates through a deformable, elastic medium i.e. sound - cannot travel through outer space Electromagnetic wave –Does not require a medium i.e. visible light, radio waves, microwaves, x rays

24. Types of Waves Pulse Wave: Single nonperiodic disturbance Periodic Wave: A wave whose source is some form of periodic motion Sine Wave: A wave whose source vibrates with simple harmonic motion –Every point vibrates up and down

25. Types of Waves Transverse wave: A wave whose particles vibrate perpendicularly to the direction of wave motion Longitudinal wave: A wave whose particles vibrate parallel to the direction of wave motion. i.e. sound Note: The distance between the adjacent crests and troughs are the same

26. Period, Frequency, and Wave speed Period is the amount of time it takes for a complete wavelength to pass a given point

27. Waves and Energy Waves carry a certain amount of energy Energy transfers from one place to another Medium remains essentially in the same place The greater the amplitude of the wave, the more energy transfered

28. Chapter 12 12.4 Wave Interactions

29. Wave Interference Waves are not matter, but displacements of matter –Two waves can occupy the same space at the same time –Forms an interference pattern Superposition: Combination of two overlapping waves

30. Constructive interference Individual displacements on the same side of the equilibrium position are added together to form a resultant wave

31. Destructive Interference Individual displacements on opposite sides of the equilibrium position are added together to form the resultant wave

32. Reflection When a wave encounters a boundary, it is reflected –If it is a free boundary/reflective surface the wave is reflected unchanged –If it is a fixed boundary, the wave is reflected and inverted

33. Standing Waves A wave pattern that results when two waves of the same frequency travel in opposite directions and interfere –Nodes: point in standing wave that always undergoes complete destructive interference and is stationary –Antinode: Point in standing wave, halfway between two nodes, with largest amplitude