Chapter 11 Vibrations and Waves Periodic Motion When

Chapter 11: Vibrations and Waves

Periodic Motion • When a vibration (aka oscillation) repeats itself back and forth over the same path the motion is called periodic. • One of the most common examples of this is an object bouncing at the end of a spring, so we will look at this example closely.

Spring – Mass Systems • Assume a massless spring attached sideways with an object of mass m attached to one end. • All springs will have a relaxed length where the mass will just sit still, this is called the equilibrium position.

Choices • There can be 3 possible scenarios. – The mass could be at the equilibrium position. – The mass could be between the equilibrium position and the other end of the spring (the spring is squished). – The mass could be beyond the equilibrium position (the spring is stretched).

You’re making me uncomfortable • A spring hates to be away from its equilibrium position. • If moved away from its EP, a spring will try to move back to it. • Remember chapter 6, springs create a force equal to spring constant times displacement. F = -kx

A sad truth • So let’s say we have a mass on a spring and we squish it in a little bit then let go. • The spring puts a force on the mass in the opposite direction we pushed it, trying to get back to EP. • The spring over shoots the EP and exerts a force pulling the mass back to EP, but it over shoots it again!

Warning: Vocab • The distance we move the mass away from EP is called the displacement. • The maximum displacement is called the amplitude. • One complete back and forth motion is called a cycle. • Period is the time per cycle and frequency is the number of cycles per time.

Simple Harmonic Motion • Any vibrating system for which the restoring force is directly proportional to the negative of the displacement is said to exhibit simple harmonic motion. – Simple: force is caused only by displacement, it’s not rocket propelled. – Harmonic: continuous back and forth movement. – Motion: duh

Simple Harmonic Oscillator • A system that creates simple harmonic motion is called a simple harmonic oscillator. • Today we examine the energies involved in such a system. • Relax, it’s just a review of chapter 6.

Getting things started • The simplest simple harmonic oscillator is our mass and spring again. • To get the system bouncing we first need to squish or stretch the spring. • When we do that we give the spring potential energy, remember PE = 1/2 kx 2 (chapter 6)

The return of conservation of energy • In chapter 6 we learned that the total energy of a system, E, is always equal to the kinetic energy plus the potential energy. • E = KE + PE • E = 1/2 mv 2 + 1/2 kx 2

Extreme Measures • It is simplest to look at the extreme points of the system to find its total E. • At the maximum displacement (called amplitude, A) all of E is PE E = 1/2 m(0)2 + 1/2 k. A 2 = 1/2 k. A 2 • At the equilibrium position, EP, all of E is KE E = 1/2 mv 02 + 1/2 k(0)2 = 1/2 mv 02 • Note: in this chapter v 0 is the MAX velocity.

The Period of Simple Harmonic Motion • V 0 = • Solving for T gives us T = • Remember 1/2 k. A 2 = 1/2 mv 2, so A/v 0 = √(m/k) • So T =

Frequency of SHM • Because f = 1/T, • f=

Position as a function of time • How can we figure out the distance our object is from the equilibrium point at any given time? • x = A cos ωt • x = A cos 2πft • x = A cos (2πt / T)

Velocity and acceleration as a function of time • v = -v 0 sin(2πt / T) • a = -(k. A/m) cos(2πft) = -a 0 cos(2πt / T)

Wave Motion • In Chapters 11 and 12 we are only concerned with one family of waves, mechanical waves. • Mechanical waves are waves created by mechanical forces – Shaking a slinky – An earthquake – A car going over a bump

A common misconception • Many people think that waves carry matter like surfing. • This is NOT true • A wave is energy, a chain reaction.

Types of waves • Transverse wave – a wave that travels perpendicular to the vibrations • Longitudinal wave – a wave that travels parallel to the vibrations – Aka density wave or pressure wave

Vocab • Pulse – a single wave bump. (What you made with the slinkies) • Continuous/Periodic wave – when the force making the wave is a vibration. (What you did when you shook the slinky constantly) • Amplitude – the max height of the wave

Even more vocab • Wavelength (λ) the distance from peak to peak. • Frequency (f) – the number of peaks per unit time. • Wave Velocity (v) – the velocity that wave peaks move.

The speed of waves • • What is speed? For a wave, what is distance? Time? So v = λ / T Because T = 1/f, we can write v = λf speed of a wave = frequency x wavelength

Speed Limit • Remember the slinky lab? • What happened as frequency increased? • The speed of a mechanical wave for any given medium is fixed.

Superposition of Waves • Last time we learned different sound waves in the same medium travel at the same speed. • So two different sounds played at the same time and distance reaches your ear at the same time. • How can 2 waves be in the same place at the same time?

Superposition of Waves 2 • Waves are not matter, they are displacements of matter. • They can occupy the same space. • The combination of two overlapping waves is called superposition.

Interference • When two waves overlap they have an effect on each other. • This effect can be observed by studying the interference pattern of the waves. • We will first look at the two extreme cases of interference.

Constructive Interference • Superposition principle – the amplitude of the resulting wave can be found by adding the amplitudes of each wave. • When the displacement is on the same side for each, the sign is the same and the wave is bigger. This is constructive interference

Destructive Interference • When the displacement of two overlapping waves are in opposite directions their signs are different. • So, when added they produce a smaller wave. This is destructive interference. • When 2 equal and opposite waves overlap, their sum is zero. This is called complete destructive interference.

Reflection of waves • If you shook your slinky hard enough you saw it start to come back. • If waves are not matter then how do they bounce off things?

Newton’s Third strikes again • Picture a string tied to a pole so that the knot can move freely • As the pulse travels down the string the knot moves up. • Tension pulls the knot back down creating a disturbance in the rope. • This creates a new wave back in the same direction.

If the end was fixed • If the string fixed to the pole instead how would it change things? • When the pulse hits the pole it wants to move up, but the pole exerts a downward force. • This force creates a wave that is in the opposite displacement of the original wave.

Standing waves • In the last slide, what would happen if the string was moved up and down not just once but constantly? • This is what is known as a standing wave

Nodes and Antinodes • Nodes (N)- places where complete destructive interference occurs. • Antinodes (A) – the midway point between two nodes. The amplitude is highest at this point.

More on standing waves • Only particular frequencies, and therefore, particular wavelengths can produce standing waves. • The wavelength of a standing wave depends on the length of the string.

Standing Waves • Yesterday we ended class by observing the standing wave. • The points of destructive interference, where the string stays still, are called nodes. • The points of constructive interference, where the string reaches its max height, are called antinodes.

All Natural • A standing wave can occur at more than one frequency. • However, they can only occur at specific frequencies. • The frequencies that do create standing waves are called the natural frequencies or resonant frequencies of the string.

Harmonics • The wavelengths of the natural frequencies depend on the length of the string itself. • The lowest frequency, called the fundamental frequency, has a wavelength given by the following: L = ½λ 1

Overtones • The other natural frequencies are called overtones. • The wavelengths of each overtone can be found using the following: L = nλn / 2 • Or λn = 2 L / n

Frequency • From v = λf we know f = v/ λ • So the frequency of each harmonic can be found using: fn = v / λn = nv / 2 L • Finally, v =√(FT / (m/L)) • With these tools we can know exactly how long to make certain strings so that they make specific noises. This is what makes music possible!