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# Physics 11: Vibrations and Waves - PowerPoint PPT Presentation

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

Christopher Chui

Physics 11: Vibrations and Waves - Christopher Chui

• 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

• 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 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 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

• 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

• 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

• 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

• 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

• 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

• 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

• 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