Vibrations and Waves Chapter 12

12.1 – Simple Harmonic Motion

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

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 http://www.ngsir.netfirms.com/englishhtm/SpringSHM.htm

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.

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

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?

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

12.2 – Measuring simple harmonic motion

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)

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)

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)

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?

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

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 0.840 s. For the first few seconds, the vibration approximates simple harmonic motion. Find the spring constant of a single spring.

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

12.3 – Properties of Waves

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

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

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

Transverse Wave Longitudinal Wave

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

The Nature of Waves Video 2:20

12.4 – Wave Interactions

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

Wave Interference Demo

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

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

Ruben's Tube Video 1:57