Oscillations About Equilibrium

7.1 Periodic Motion

Periodic Motion – repeat, same time, same path Period (T) – time required for one complete cycle (seconds) or seconds/cycle Frequency ( f ) – the number of oscillations per second (s -1 or hertz) 7.2 Simple Harmonic Motion

A form of Periodic Motion Simple Harmonic Motion A restoring force is applied proportional to the distance from equilibrium So Hooke’s Law 7.2 Simple Harmonic Motion

If a graph of simple harmonic motion is created And spread out over time We get a wave pattern Amplitude – maximum displacement 7.2 Simple Harmonic Motion

The mathematical relationship between displacement and position is When given in this form, you can determine the Amplitude, and Period of a particle undergoing simple harmonic motion 7.2 Simple Harmonic Motion

7.3 The Period of a Mass on a Spring

The period of a spring is given by the equation A larger mass would have greater inertia – longer period A larger spring constant would produce more acceleration, so a shorter period The period is independent of amplitude 7.3 The Period of a Mass on a Spring

If a spring is hung vertically Only the equilibrium position changes At equilibrium The weight is a constant force, so the motion is still dependant on just the force of the spring 7.3 The Period of a Mass on a Spring

7.4 Energy Conservation in Oscillatory Motion

If we ignore friction, then he energy of a spring moving horizontally is The maximum value for energy is 7.4 Energy Conservation in Oscillatory Motion

7.5 The Pendulum

A simple Pendulum The potential energy is So potential energy is zero at equilibrium (like SHM) 7.5 The Pendulum L Lcos  L-Lcos  

The period of a pendulum is given as Independent of the mass of the bob 7.5 The Pendulum

Restoring Force Forces Components A pendulum does not act as a Simple Harmonic Oscillator, but at small angles (<30 o ) it approximates SHM 7.5 The Pendulum W T mgsin  mgcos 

7.6 Damped Oscillations

7.6 Damped Oscillation The amplitude of a real oscillating object will decrease with time – called damping Underdamped – takes several swing before coming to rest (above)

7.6 Damped Oscillation Overdamped – takes a long time to reach equilibrium Critical damping – equalibrium reached in the shortest time

7.7 Driven Oscillations and Resonance

Natural Frequency – depends on the variables (m,k or L,g) of the object Forced Vibrations – caused by an external force

7.7 Driven Oscillations and Resonance Resonant Frequency – the natural vibrating frequency of a system Resonance – when the external frequency is near the natural frequency and damping is small Tacoma Narrow Bridge

7.8 Types of Waves

Mechanical Waves – travels through a medium The wave travels through the medium, but the medium undergoes simple harmonic motion Wave motion Particle motion

7.8 Types of Waves Waves transfer energy, not particles A single bump of a wave is called a pulse A wave is formed when a force is applied to one end Each successive particle is moved by the one next to it

7.8 Types of Waves Parts of a wave Transverse wave – particle motion perpenduclar to wave motion Wavelength ( ) measured in meters Frequency ( f ) measured in Hertz (Hz) Wave Velocity ( v ) meters/second

7.8 Types of Waves Longitudinal (Compressional) Wave Particles move parallel to the direction of wave motion Rarefaction – where particles are spread out Compression – particles are close

7.8 Types of Waves Earthquakes S wave – Transverse P wave – Longitudinal Surface Waves – can travel along the boundary Notice the circular motion of the particles

7.9 Reflection and Transmission of Waves

When a wave comes to a boundary (change in medium) at least some of the wave is reflected The type of reflection depends on if the boundary is fixed (hard) - inverted

7.9 Reflection and Transmission of Waves When a wave comes to a boundary (change in medium) at least some of the wave is reflected Or movable (soft) – in phase

7.9 Reflection and Transmission of Waves For two or three dimensional we think in terms of wave fronts A line drawn perpendicular to the wave front is called a ray When the waves get far from their source and are nearly straight, they are called plane waves

7.9 Reflection and Transmission of Waves Law of Reflection – the angle of reflection equals the angle of incidence Angles are always measured from the normal

7.9 Reflection and Transmission of Waves Law of Reflection – the angle of reflection equals the angle of incidence Angles are always measured from the normal

7.10 Characteristics of Sound

Sound is a longitudinal wave Caused by the vibration of a medium The speed of sound depends on the medium it is in, and the temperature For air, it is calculated as

7.10 Characteristics of Sound Loudness – sensation of intensity Pitch – sensation of frequency Range of human hearing – 20Hz to 20,000 Hz ultrasonic – higher than human hearing dogs hear to 50,000 Hz, bats to 100,000 Hz infrasonic – lower than human hearing

7.10 Characteristics of Sound Often called pressure waves Vibration produces areas of higher pressure These changes in pressure are recorded by the ear drum

7.11 Intensity of Sound

Loudness – sensation Relative to surrounding and intensity Intensity – power per unit area Humans can detect intensities as low as W/m 2 The threshold of pain is 1 W/m 2

7.11 Intensity of Sound Sound intensity is usually measured in decibels (dB) Sound level is given as I – intensity of the sound I 0 – threshold of hearing ( W/m 2 )  – sound level in dB Some common relative intensities Source of SoundSound Level (dB) Jet Plane at 30 m140 Threshold of Pain120 Loud Rock Concert120 Siren at 30 m100 Auto Interior at 90 km/h75 Busy Street Traffic70 Conversation at 0.50 m65 Quiet Radio40 Whisper20 Rustle of Leaves10 Threshold of Hearing0

7.12 The Ear

Steps in sound transmission

7.13 Sources of Sound: Strings and Air Columns

7.13 Sources of Sound Vibrations in strings Fundamental frequency Next Harmonic

7.13 Sources of Sound Vibrations in strings Next Harmonic Strings produce all harmonics – all whole number multiples of the fundamental frequency

7.13 Sources of Sound Vibrations in an open ended tube (both ends) Fundamental frequency Next Harmonic

7.13 Sources of Sound Vibrations in open ended tubes Next Harmonic Open ended tubes produce all harmonics – all whole number multiples of the fundamental frequency

7.13 Sources of Sound Vibrations in an closed end tube (one end) Fundamental frequency Next Harmonic

7.13 Sources of Sound Vibrations in open ended tubes Next Harmonic Closed end tubes produce only odd harmonics

7.14 Interference of Sound Waves: Beats

If waves are produced by two identical sources A pattern of constructive and destructive interference forms Applet

7.15 The Doppler Effect

Doppler Effect – the change in pitch due to the relative motion between a source of sound and the receiver Applies to all wave phenomena Objects moving toward you have a higher apparent frequency Objects moving away have a lower apparent frequency Doppler Effect Light Doppler

7.15 The Doppler Effect If an object is stationary the equation for the wave velocity is Sound waves travel outward evenly in all directions If the object moves toward the observed, the waves travel at the same velocity, but each new vibration is created closer to the observer Doppler Applet

7.15 The Doppler Effect The general equation is The values of V o (speed of observer) and V s (speed of source) is positive when they approach each other Radar Gun

7.16 Interference

Interference – two waves pass through the same region of space at the same time The waves pass through each other Principle of Superposition – at the point where the waves meet the displacement of the medium is the algebraic sum of their separate displacements

7.16 Interference Phase – relative position of the wave crests If the two waves are “in phase” Constructive interference If the two waves are “out of phase” Destructive Interference

7.16 Interference For a wave (instead of a single phase) Interference is calculated by adding amplitude In real time this looks like