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##### Physics 11: Vibrations and Waves

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**Physics 11: Vibrations and Waves**Christopher Chui 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 • 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**• 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**• 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**• 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**• 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**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**• 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**• 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**• 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**• 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**• 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