Vibrations and Waves Chapter 12

12.1 – Simple Harmonic Motion

Remember…

Elastic Potential Energy (PE e )

energy stored in a stretched or is the compressed elastic object

Gravitational Potential Energy (PE g )

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/Spring SHM.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)

equilibrium.

is the distance from (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: F elastic = -(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 g = 9.81

x = -0.020m

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 = 2 π√(L/g)

Period = 2 π 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 = 2 π√(m/k)

Period = 2 π 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 = 2 π√(L/g)

12 = 2 π√(L/9.81) 144 = 4 π 2 L/9.81

1412.64 = 4 π 2 L 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 = 2 π√(m/k) K=(4 π 2 m)/T 2 K= (4 π 2 (357 kg))/(0.840 s) 2 k= 2.00*10 4 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 =

f λ 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.

v =

f λ 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 At a fixed boundary, waves are reflected and inverted Animations courtesy of Dr. Dan Russell, Kettering University

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