1 / 12

Physics 11: Vibrations and Waves

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

rebecca
Télécharger la présentation

Physics 11: Vibrations and Waves

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Physics 11: Vibrations and Waves Christopher Chui Physics 11: Vibrations and Waves - Christopher Chui

  2. 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

  3. 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

  4. 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

  5. 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

  6. 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

  7. 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

  8. 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

  9. 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

  10. 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

  11. 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

  12. 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

More Related