Vibration and Waves AP Physics Chapter 11

11.1 Simple Harmonic Motion Vibration and Waves 11.1 Simple Harmonic Motion

11.1 Simple Harmonic Motion Periodic motion – when an object vibrates over the same pathway, with each vibration taking the same amount of time Equilibrium position – the position of the mass when no force is exerted on it 11.1

11.1 Simple Harmonic Motion If the spring is stretched from equilibrium, a force acts so the object is pushed back toward equilibrium Restoring Force Proportional to the displacement (x) Called Hooke’s Law 11.1

11.1 Simple Harmonic Motion Any vibrating system for which the restoring force is directly proportional to the negative of the displacement (F = -kx) exhibits Simple Harmonic Motion (SHM)

11.1 Simple Harmonic Motion Amplitude (A) – maximum distance from equilibrium Period (T) – time for one complete cycle Frequency (f) – number of vibrations per second Amplitude 11.1

11.1 Simple Harmonic Motion A vertical spring follows the same pattern The equilibrium positions is just shifted by gravity 11.1

11.2 Energy in a Simple Harmonic Oscillator Vibration and Waves 11.2 Energy in a Simple Harmonic Oscillator

11.2 Energy in a Simple Harmonic Oscillator Review – energy of a spring So the total mechanical energy of a spring (assuming no energy loss) is At maximum amplitude then So E is proportional to the square of the amplitude 11.2

11.2 Energy in a Simple Harmonic Oscillator Velocity as a function of position Since the maximum velocity is when A=0 Factor the top equation for A2, then combine with the bottom equation Take square root 11.2

11.3 The Period and Sinusoidal Nature of SHM Vibration and Waves 11.3 The Period and Sinusoidal Nature of SHM

11.3 The Period and Sinusoidal Nature of SHM The Period of an object undergoing SHM is independent of the amplitude Imagine an object traveling in a circular pathway If we look at the motion in just the x axis, the motion is analogous to SHM 11.3

11.3 The Period and Sinusoidal Nature of SHM As the ball moves the displacement in the x changes The radius is the amplitude The velocity is tangent to the circle Now looking at Components If we put the angles into the triangle, we can see similar triangles Vmax q V A q 11.3

11.3 The Period and Sinusoidal Nature of SHM So the opposite/hypotenuse is a constant This can be rewritten This is the same as the equation for velocity of an object in SHM Vmax q V A q 11.3

11.3 The Period and Sinusoidal Nature of SHM The period would be the time for one complete revolution The radius is the same as the amplitude, and the time for one revolution is the period 11.3

11.3 The Period and Sinusoidal Nature of SHM Using the previous relationship between maximum velocity and amplitude Substitute in the top equation 11.3

11.3 The Period and Sinusoidal Nature of SHM Position as a function of time x displacement is You don’t know this, but Where 𝑓 is the frequency So A q 11.3

Vibration and Waves 11.4 The Simple Pendulum

Cord is massless (or very small) Mass is concentrated in small volume 11.4 The Simple Pendulum Simple Pendulum – mass suspended from a cord Cord is massless (or very small) Mass is concentrated in small volume 11.4

Looking at a diagram of a pendulum Two forces act on the it Weight 11.4 The Simple Pendulum Looking at a diagram of a pendulum Two forces act on the it Weight Tension The motion of the bob is at a tangent to the arc T W 11.4

The displacement of the bob is given by x Using the triangle 11.4 The Simple Pendulum The displacement of the bob is given by x Using the triangle For a complete circle (360o) Then for our arc it would be q L Ly T W x 11.4

The component of force in the direction of motion is 11.4 The Simple Pendulum The component of force in the direction of motion is But at small angle And So q L Ly T mgsinq W x 11.4

Usual standard is below 30o 11.4 The Simple Pendulum Usual standard is below 30o We can then take the equation and reason that 360o=2p rad (we didn’t study angular motion, so take my word for it) 11.4

We can now combine the equation for period 11.4 The Simple Pendulum We can now combine the equation for period Mass does not appear in this equation The period is independent of mass 11.4

11.5 Damped Harmonic Motion Vibration and Waves 11.5 Damped Harmonic Motion

11.5 Damped Harmonic Motion The amplitude of a real oscillating object will decrease with time – called damping Underdamped – takes several swing before coming to rest (above) 11.5

11.5 Damped Harmonic Motion Overdamped – takes a long time to reach equilibrium Critical damping – equalibrium reached in the shortest time 11.5

11.6 Forced Vibrations; Resonance Vibration and Waves 11.6 Forced Vibrations; Resonance

11.6 Forced Vibrations; Resonance Natural Frequency – depends on the variables (m,k or L,g) of the object Forced Vibrations – caused by an external force 11.6

11.6 Forced Vibrations; 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 11.6

Vibration and Waves 11.7 Wave Motion

Mechanical Waves – travels through a medium 11.7 Wave Motion Mechanical Waves – travels through a medium The wave travels through the medium, but the medium undergoes simple harmonic motion Wave motion Particle motion 11.7

Waves transfer energy, not particles 11.7 Wave Motion 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 Tsunami 11.7

perpenduclar to wave motion Wavelength (l) measured in meters Parts of a wave Transverse wave – particle motion perpenduclar to wave motion Wavelength (l) measured in meters Frequency (f) measured in Hertz (Hz) Wave Velocity (v) meters/second 11.7

Longitudinal (Compressional) Wave Particles move parallel to the 11.7 Wave Motion Longitudinal (Compressional) Wave Particles move parallel to the direction of wave motion Rarefaction – where particles are spread out Compression – particles are close 11.7

Surface Waves – can travel along the boundary 11.7 Wave Motion Earthquakes S wave – Transverse P wave – Longitudinal Surface Waves – can travel along the boundary Notice the circular motion of the particles 11.7

11.9 Energy Transported by Waves Vibration and Waves 11.9 Energy Transported by Waves

11.9 Energy Transported by Waves Energy for a particle undergoing simple harmonic motion is Intensity (I) power across a unit area perpendicular to the direction of energy flow So 11.9

11.11 Reflection and Transmission of Waves Vibration and Waves 11.11 Reflection and Transmission of Waves

11.11 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 11.11

11.11 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 11.11

11.11 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 11.11

11.11 Reflection and Transmission of Waves Law of Reflection – the angle of reflection equals the angle of incidence Angles are always measured from the normal 11.11

11.12 Interference; Principle of Superposition Vibration and Waves 11.12 Interference; Principle of Superposition

11.12 Interference; Principle of Superposition 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 11.12

11.12 Interference; Principle of Superposition 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 11.12

11.12 Interference; Principle of Superposition For a wave (instead of a single phase) Interference is calculated by adding amplitude In real time this looks like 11.12

11.13 Standing Waves; Resonance Vibration and Waves 11.13 Standing Waves; Resonance

11.13 Standing Waves; Resonance In a specific case of interference a standing wave is produced The areas with complete constructive interference are called loops or antinodes (AN) The areas with complete destructive interference are called nodes (N) 11.13

11.13 Standing Waves; Resonance Standing waves occur at the natural or resonant frequency of the medium In this case, called the first harmonic, the wavelength is twice the length of the medium The frequency of is called the fundamental frequency 11.13

11.13 Standing Waves; Resonance The second harmonic is the next standing wave formed Then the third harmonic would be 11.13

11.13 Standing Waves; Resonance The basic form for the wavelength of harmonics is Each resonant frequency, is an integer multiple of the fundamental frequency Overtone – all the frequencies above the fundamental The first overtone is the second harmonic Harmonic Applet 11.13