* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download damped and driven oscillations, waves
Hunting oscillation wikipedia , lookup
Photoelectric effect wikipedia , lookup
Quantum tunnelling wikipedia , lookup
Old quantum theory wikipedia , lookup
Introduction to quantum mechanics wikipedia , lookup
Classical central-force problem wikipedia , lookup
Density of states wikipedia , lookup
Coherence (physics) wikipedia , lookup
Double-slit experiment wikipedia , lookup
Electromagnetic spectrum wikipedia , lookup
Wave function wikipedia , lookup
Shear wave splitting wikipedia , lookup
Photon polarization wikipedia , lookup
Stokes wave wikipedia , lookup
Wave packet wikipedia , lookup
Surface wave inversion wikipedia , lookup
Matter wave wikipedia , lookup
Theoretical and experimental justification for the Schrödinger equation wikipedia , lookup
Damped and Forced SHM Physics 202 Professor Vogel (Professor Carkner’s notes, ed) Lecture 4 Damped SHM Consider a system of SHM where friction is present The mass will slow down over time The damping force is usually proportional to the velocity The faster it is moving, the more energy it loses If the damping force is represented by Fd = -bv Where b is the damping constant Then, x = xmcos(wt+f) e(-bt/2m) e(-bt/2m) is called the damping factor and tells you by what factor the amplitude has dropped for a given time or: x’m = xm e(-bt/2m) Energy and Frequency The energy of the system is: E = ½kxm2 e(-bt/m) The energy and amplitude will decay with time exponentially The period will change as well: w’ = [(k/m) - (b2/4m2)]½ For small values of b: w’ ~ w Exponential Damping Damped Systems All real systems of SHM experience damping Most damping comes from 2 sources: Air resistance Example: the slowing of a pendulum Energy dissipation Example: heat generated by a spring Lost energy usually goes into heat Damping Forced Oscillations If this force is applied periodically then you have If you apply an additional force to a SHM system you create forced oscillations Example: pushing a swing 2 frequencies for the system w = the natural frequency of the system wd = the frequency of the driving force The amplitude of the motion will increase the fastest when w=wd Resonance The condition where w=wd is called resonance Resonance occurs when you apply maximum driving force at the point where the system is experiencing maximum natural force Example: pushing a swing when it is all the way up All structures have natural frequencies When the structures are driven at these natural frequencies large amplitude vibrations can occur What is a Wave? If you wish to move something (energy, information etc.) from one place to another you can use a particle or a wave Example: transmitting energy, A bullet will move energy from one place to another by physically moving itself A sound wave can also transmit energy but the original packet of air undergoes no net displacement Transverse and Longitudinal Transverse waves are waves where the oscillations are perpendicular to the direction of travel Examples: waves on a string, ocean waves Sometimes called shear waves Longitudinal waves are waves where the oscillations are parallel to the direction of travel Examples: slinky, sound waves Sometimes called pressure waves Transverse Wave Longitudinal Wave Waves and Medium Waves travel through a medium (string, air etc.) The wave has a net displacement but the medium does not Each individual particle only moves up or down or side to side with simple harmonic motion This only holds true for mechanical waves Photons, electrons and other particles can travel as a wave with no medium (see Chapter 33) Wave Properties Consider a transverse wave traveling in the x direction and oscillating in the y direction The y position is a function of both time and x position and can be represented as: y(x,t) = ym sin (kx-wt) Where: ym = amplitude k = angular wave number w = angular frequency Wavelength and Number A wavelength (l) is the distance along the x-axis for one complete cycle of the wave One wavelength must include a maximum and a minimum and cross the x-axis twice We will often refer to the angular wave number k, k=2p/l Period and Frequency Period is the time for one wavelength to pass a point Frequency is the number of oscillations (wavelengths) per second (f=1/T) We will again use the angular frequency w, w=2p/T The quantity (kx-wt) is called the phase of the wave Speed of a Wave Our equation for the wave, tells us the “updown” position of some part of the medium y(x,t) = ym sin (kx-wt) But we want to know how fast the waveform moves along the x axis: v=dx/dt We need an expression for x in terms of t If we wish to discuss the wave form (not the medium) then y = constant and: kx-wt = constant e.g. the peak of the wave is when (kx-wt) = p/2 we want to know how fast the peak moves Wave Speed Velocity We can take the derivative of this expression w.r.t time (t): k(dx/dt) - w = 0 (dx/dt) = w/k = v Since w = 2pf and k = 2p/l v = w/k = 2pfl/2p v = lf Thus, the speed of the wave is the number of wavelengths per second times the length of each i.e. v is the velocity of the wave form