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Physics 11: Vibrations and Waves. Christopher Chui. Simple Harmonic Motion (SHM). Any spring has a natural length at which it exerts no force on the mass is called equilibrium If stretched, the restoring force F = -kx, called SHM The stretched distance, x, is displacement

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Physics 11 vibrations and waves l.jpg

Physics 11: Vibrations and Waves

Christopher Chui

Physics 11: Vibrations and Waves - Christopher Chui


Simple harmonic motion shm l.jpg
Simple Harmonic Motion (SHM)

  • Any spring has a natural length at which it exerts no force on the mass is called equilibrium

  • If stretched, the restoring force F = -kx, called SHM

  • The stretched distance, x, is displacement

  • The max displacement is called amplitude, A

  • One cycle is one complete to-and-fro (-A to +A) motion

  • Period, T, is the time for one complete cycle

  • Frequency, f, is the number of complete cycles in one second. T = 1/f and f = 1/T

Physics 11: Vibrations and Waves - Christopher Chui


Energy in sho l.jpg
Energy in SHO

  • PE = ½ kx2 k is called the spring constant

  • Total mechanical energy, E = ½ mv2 + ½ kx2

  • At the extreme points, E = ½ kA2

  • At the equilibrium point, E = ½ mvo2 vo is max

  • Using conservation of energy, we find at any time, the velocity v = +- vo [sqrt(1 – x2/A2)]

Physics 11: Vibrations and Waves - Christopher Chui


The period and sinusoid of shm l.jpg
The Period and Sinusoid of SHM

  • The period does not depend on the amplitude

  • For a revolving object making one revolution, vo = circumference / time = 2pA / T = 2pAf

  • Since ½ kA2 = ½ mvo2, T = 2p sqrt(m/k)

  • Since f=1/T, f = 1/(2p) sqrt(k/m)

  • x = Acos q = Acos wt = Acos 2pft = Acos 2pt/T

  • v = -vo sin 2pft = -vo sin 2pt/T

  • A = F/m = -kx/m = -[kA/m] cos 2pft = -aocos2pft

Physics 11: Vibrations and Waves - Christopher Chui


The simple pendulum of length l l.jpg
The Simple Pendulum of length L

  • The restoring force, F = - mg sin q

  • For small angles, sin q is approx = to q

  • F = -mg q = -mg x/L = -kx, where k = mg/L

  • The period, T = 2 p sqrt (L/g)

  • The frequency, f = 1/T = 1/(2 p) sqrt (g/L)

Physics 11: Vibrations and Waves - Christopher Chui


Damped harmonic motion l.jpg
Damped Harmonic Motion

  • Automobile spring and shock absorbers provide damping so that the car won’t bounce up and down

  • Overdamped takes a long time to reach equilibrium

  • Underdamped takes several bounces before coming to rest

  • Critical damping reaches equilibrium the fastest

Physics 11: Vibrations and Waves - Christopher Chui


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Forced Vibrations and Resonance

  • A system with a natural frequency may have a force applied to it. This is a forced vibration

  • If the applied force = its natural frequency, then we have resonance. This freq is resonance freq. This will lead to resonant collapse

Physics 11: Vibrations and Waves - Christopher Chui


Wave motion l.jpg
Wave Motion

  • Waves are moving oscillations, not carrying matter along

  • A simple wave bump is a wave pulse

  • A continuous or periodic wave has at its source a continuous and oscillating disturbance

  • The amplitude is the max height of a crest

  • The distance between two consecutive crests is called the wavelength, l

  • The frequency, f, is the number of complete cycles

  • The wave velocity, v = lf, is the velocity at which wave crests move, not the velocity of the particle

  • For small amplitude, v = sqrt [FT/(m/L)] , m/L: mass/length

Physics 11: Vibrations and Waves - Christopher Chui


Transverse and longitudinal waves l.jpg
Transverse and Longitudinal Waves

  • Particles vibrate up and down = transverse wave

  • Particles vibrate in the same direction = longitudinal wave, resulting in compression and expansion

  • The velocity of longitudinal wave = sqrt (elastic force factor / inertia force factor)=sqrt (E/ r)

  • For liquid or gas, v = sqrt (B/ r), r is the density

Physics 11: Vibrations and Waves - Christopher Chui


Energy of waves l.jpg
Energy of Waves

  • Wave energy is proportional to the square of amplitude

  • Intensity, I = energy/time/area = power/area

  • For a spherical wave, I = P/4pr2

  • For 2 points at r1 and r2, I2/I1 = r12 / r22

  • For wave twice as far, the amplitude is ½ as large, such that A2/A1 = r1 /r2

Physics 11: Vibrations and Waves - Christopher Chui


Reflection and interference l.jpg
Reflection and Interference

  • The law of reflection: the angle of incidence = the angle of reflection

  • Interference happens when two waves pass through the same region at the same time

  • The resultant displacement is the algebraic sum of their separate displacements

  • A crest is positive and a trough is negative

  • Superposition results in either constructive or destructive

  • 2 constructive waves are in phase; destructive waves are out of phase

Physics 11: Vibrations and Waves - Christopher Chui


Standing wave and resonance l.jpg
Standing Wave and Resonance

  • 2 traveling waves may interfere to give a large amplitude standing wave

  • The points of destructive interference are nodes

  • Points of constructive interference are antinodes

  • Frequencies at which standing waves are produced are natural freq or resonance freq

  • Only standing waves with resonant frequencies persist for long such as guitar, violin, or piano

  • The lowest frequency is the fundamental freq = 1 antinode, L = 1st harmonic = ½ l1

  • The other natural freq are overtones, multiples of fundamental frequencies, L = nln/2 n = 1, 2, 3, ...

Physics 11: Vibrations and Waves - Christopher Chui