1. Vibrations and Waves
Chapter 12

2. 12.1 – Simple Harmonic Motion

3. Remember…
Elastic Potential Energy (PEe) is the energy stored in a stretched or compressed elastic object
Gravitational Potential Energy (PEg) is the energy associated with an object due to it’s position relative to Earth

4. Useful Definitions
Periodic Motion – A repeated motion. If it is back and forth over the same path, it is called simple harmonic motion.
Examples: Wrecking ball, pendulum of clock
Simple Harmonic Motion – Vibration about an equilibrium position in which a restoring force is proportional to the displacement from equilibrium

5. Useful Definitions
A spring constant (k) is a measure of how resistant a spring is to being compressed or stretched.
(k) is always a positive number
The displacement (x) is the distance from equilibrium.
(x) can be positive or negative. In a spring-mass system, positive force means a negative displacement, and negative force means a positive displacement.

6. Hooke’s Law Hooke’s Law – for small displacements from equilibrium:
Felastic = -(kx)
Spring force = -(spring constant x displacement)
This means a stretched or compressed spring has elastic potential energy.
Example: Bow and Arrow

7. Example Problem
If a mass of 0.55 kg attached to a vertical spring stretches the spring 2.0 cm from its original equilibrium position, what is the spring constant?

8. Example Answer Given: m = 0.55 kg x = -0.020m g = 9.81 k = ?
Fg = mg = 0.55 kg x 9.81 = 5.40 N
Hooke’s Law: F = -kx
5.40 N = -k(-0.020m) k = 270 N/m

9. 12.2 – Measuring simple harmonic motion

10. Useful Definitions
Amplitude – the maximum angular displacement from equilibrium.
Period – the time it takes to execute a complete cycle of motion
Symbol = T SI Unit = second (s)
Frequency – the number of cycles or vibrations per unit of time
Symbol = f SI Unit = hertz (Hz)

11. Formulas - Pendulums T = 1/f or f = 1/T
The period of a pendulum depends on the string length and free-fall acceleration (g)
T = 2π√(L/g)
Period = 2π x square root of (length divided by free-fall acceleration)

12. Formulas – Mass-spring systems
Period of a mass-spring system depends on mass and spring constant
A heavier mass has a greater period, thus as mass increases, the period of vibration increases.
T = 2π√(m/k)
Period = 2π x the square root of (mass divided by spring constant)

13. Example Problem- Pendulum
You need to know the height of a tower, but darkness obscures the ceiling. You note that a pendulum extending from the ceiling almost touches the floor and that its period is 12s. How tall is the tower?

14. Example Answer Given: T = 12 s g = 9.81 L = ? T = 2π√(L/g)
35.8 m = L

15. Example Problem- Mass-Spring
The body of a 1275 kg car is supported in a frame by four springs. Two people riding in the car have a combined mass of 153 kg. When driven over a pothole in the road, the frame vibrates with a period of s. For the first few seconds, the vibration approximates simple harmonic motion. Find the spring constant of a single spring.

16. Example answer Total mass of car + people = 1428 kg
Mass on 1 tire: 1428 kg/4= 357 kg
T= s
T = 2π√(m/k)
K=(4π2m)/T2
K= (4π2(357 kg))/(0.840 s)2
k= 2.00*104 N/m

17. 12.3 – Properties of Waves

18. Useful Definitions
Crest: the highest point above the equilibrium position
Trough: the lowest point below the equilibrium position
Wavelength λ : the distance between two adjacent similar points of the wave

19. Wave Motion A wave is the motion of a disturbance.
Medium: the material through which a disturbance travels
Mechanical waves: a wave that requires a medium to travel through
Electromagnetic waves: do not require a medium to travel through

20. Wave Types Pulse wave: a single, non-periodic disturbance
Periodic wave: a wave whose source is some form of periodic motion
When the periodic motion is simple harmonic motion, then the wave is a SINE WAVE (a type of periodic wave)
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

22. Transverse Wave
Longitudinal Wave

23. Speed of a Wave Speed of a wave= frequency x wavelength v = fλ
Example Problem:
The piano string tuned to middle C vibrates with a frequency of 264 Hz. Assuming the speed of sound in air is 343 m/s, find the wavelength of the sound waves produced by the string.
343 m/s = (264 Hz)(λ)
1.30 m = λ

24. The Nature of Waves Video 2:20

25. 12.4 – Wave Interactions

26. Constructive vs Destructive Interference
Constructive Interference: individual displacements on the same side of the equilibrium position are added together to form the resultant wave
Destructive Interference: individual displacements on the opposite sides of the equilibrium position are added together to form the resultant wave

27. Wave Interference Demo

28. When Waves Reach a Boundary…
At a free boundary, waves are reflected
Animations courtesy of Dr. Dan Russell, Kettering University
At a fixed boundary, waves are reflected and inverted

29. Standing Waves
Standing wave: a wave pattern that results when two waves of the same frequency, wavelength, and amplitude travel in opposite directions and interfere
Node: a point in a standing wave that always undergoes complete destructive interference and therefore is stationary
Antinode: a point in a standing wave, halfway between two nodes, at which the largest amplitude occurs

31. Ruben's Tube Video 1:57