Oscillations and Waves Separation between two atoms in a molecule r(t) What’s the difference? Oscillations involve a discrete set of quantities that r(t) r(t) vary in time (usually periodically). Examples: pendula, vibrations of individual molecules, firefly lights, currents in circuits. tt Waves involve continuous quantities the vary in both space and time. (variation may be periodic or not) Examples: light waves, sound waves, elastic waves, surface waves, electrochemical waves on neurons 1/23/13 1 Physics 132 Learning about Oscillations and waves • Why to learn it • What to learn • How the ear senses sound • Sound itself • Brain waves • Heart contraction waves • Molecule oscillations 1/23/13 • How to describe oscillations mathematically (sin, cos) • How to think about waves • Resonances 2 Physics 132 • Heart beat • Ventricular Fibrillation • http://www.youtube.co m/watch?annotation_i d=annotation_611436& http://www.youtube.com/watch? v=riUAFkV7HCU feature=iv&src_vid=Pes 9O5z8efk&v=uR4t__B-‐ Zwg 3 Physics 132 What are the key ingredients for oscillatory motion? 1. Motion given by Newton’s laws a = F/m 2. Force F is derived from a potential with a local minimum. dU(r) F(r) = − ; −k(r − req ) dr k U(r) ; (r − req )2 + U(req ) 2 Potential Energy - U ( r) rr rreq eq Separation - r Local Local m minimum inimum rreq eq rr11 rr rr22 Separation - r EE r(t) = req + A cos(ω 0 (t − t 0 )) Local Local m minimum inimum r(t) r(t) ω0 = rr22 AA rreq eq k m A = (r2 − req ) rr11 tt t0 = natural frequency of oscillation Amplitude of Oscillation Any time r(t)=r2 Model system: Mass on a Spring Consider a cart of mass m attached to a light (mass of spring << m) spring. Choose the coordinate system so that when the cart is at 0 the spring it at its rest length Recall the properties of an ideal spring. • When it is pulled or pushed on both ends it changes its length. • T is tension, T>0 means the spring is being stretched T = kΔl 6 Physics 132 Analyzing the forces: cart & spring 0 • What are the forces acting on the cart? x N t→c T s→c W e→c 7 What What ddo o tthe he subscripts subscripts m mean ean ?? Physics 132 A mass connected to a spring is oscillating back and forth. Consider two possibilities: (i) at some point during the oscillation the mass has v = 0 but a ≠0 (ii) at some point during the oscillation the mass has v = 0 and a = 0 . 1. 2. 3. 4. Both occur sometime during the oscillation. Neither occurs during the oscillation. Only (i) occurs. Only (ii) occurs. 8 Physics 132 Tracking the motion x F net x a v 0 x 0 0 0 x 0 x 0 0 9 0 PhysicsWhiteboard, 132 TA & LA Letʼs try it Sonic ranger Cart Air track Why do we have two springs? 10 Physics 132 • Position of the cart depends on time t • Lets call the x position of the cart: A(t) x A(t) 1/23/13 11 Physics 132 Doing the Math: The Equation of Motion x Newton’s equation for the cart is Fnet −k x(t) ⎛ k ⎞ a= = = − ⎜ ⎟ x(t) m m ⎝ m⎠ Acceleration Acceleration iis s ddefined efined d 2 x(t) a= 2 dt 12 Physics 132 Solving a differential equation • Express acceleration acceleration a as a derivative of x(t). • Verify solution x(t) = A cos(ω 0 (t − t 0 )) ω0 = k m d cosθ = − cosθ 2 dθ 2 What What ddetermines etermines AA aand nd tt00 ?? 3/14/12 ⎛ k⎞ d 2 x(t) a= = − ⎜ ⎟ x(t) 2 ⎝ m⎠ dt 13 PhysicsWhiteboard, 132 TA & LA Graphs: sin(θ) vs cos(θ) • Which is which? How can you tell? • The two functions sin and cos are derivatives of each other (slopes), but one has a minus sign. Which one? How can you tell? sin sin θθ 2/4/11 1 θ cos cos θθ 14 sin θ cos θ Physics 122 Graphs: sin(θ) vs sin(ω0t) • For angles, θ = 0 and θ = 2π are the same so you only get one cycle. What does • For time, t can go on forever changing ω0 do to this graph? so the cycles repeat. cos(θ) cos(ω0t) tt 2/4/11 15 Physics 122 Interpreting the Result • What do the various terms mean? • A is the maximum displacement — the amplitude of the oscillation. • What is ω0? If T is the period (how long it takes to go through a full oscillation) then T ω 0 t : 0 → 2π x(t) t t :0→T ω 0 T = 2π 16 ⇒ 2π ω0 = T Physics 132 If curve (A) is A cos(ω 0 t ) which curve is A cos(2ω 0 t )? 1. 2. 3. 4. 17 (A) (B) (C) None of the above. Physics 122 Which of these curves is described by A cos(ω 0 t + φ ) with φ > 0 (and φ << 2π)? 1. 2. 3. 4. 18 (A) (B) (C) None of the above. PhysicsWhiteboard, 122 TA & LA Oscillations • A typical oscillation A t t0 x ( tx)( t=) A = cos( A cos( ω 0ω( t0 t−) t 0 )) A(t) = A0 cos(ω 0 (t − t0 )) = A0 cos(ω 0t − ω 0t0 ) = A0 cos(ω 0t − φ ) Physics 132 19 Total Energy E = KE + PE rreq eq rr11 rr rr22 Separation - r EE r(t) = req + A cos(ω 0 (t − t 0 )) Local Local m minimum inimum Kinetic Potential PE = U k U(r) ; (r − req )2 + U(req ) 2 k 2 U = A cos 2 (ω (t − t 0 )) + U(req ) 2 m m dr(t) 2 2 KE = (v(t)) = ( ) 2 2 dt mω 02 2 2 KE = A sin (ω (t − t 0 )) 2 add them together Whiteboard, TA & LA k ω = m 2 0 2 k 2 m ω 0 E = U(req ) + A cos 2 (ω (t − t 0 )) + A 2 cos 2 (ω (t − t 0 )) 2 2 k 2 = U(req ) + A ⎡⎣ cos 2 (ω (t − t 0 )) + sin 2 (ω (t − t 0 )) ⎤⎦ 2 k = U(req ) + A 2 Total energy is constant The Simple Pendulum  Consider a mass m attached to a string of length L which is free to swing back and forth.  If it is displaced from its lowest position by an angle θ, Newton’s second law for the tangential component of gravity, parallel to the motion, is: 4/4/14 Physics 132 22 The Simple Pendulum If we restrict the pendulum’s oscillations to small angles (< 10°), then we may use the small angle approximation sin θ ≈ θ, where θ is measured in radians. and the angular frequency of the motion is found to be: 4/4/14 Physics 132 23Slide 14-74 Now Now aadd dd ffriction riction rreq eq rr11 rr rr22 Separation - r EE Local Local m minimum inimum r(t) r(t) γ r(t) = req + A exp[− (t − t 0 )]cos(ω 1 (t − t 0 )) 2 2 γ ω 12 = ω 02 − 4 ( ) ma = −k r − req − bv(t) d2 dr 2 r(t) = −ω 0 r(t) − γ 2 dt dt b γ = m Damped Oscillations When a mass on a spring experiences the force of the spring as given by Hooke’s Law, as well as a linear drag force of magnitude |D| = bv, the solution is: where the angular frequency is given by: Here is the angular frequency of the undamped oscillator (b = 0). 4/4/14 Physics 132 25 Damped Oscillations Position-versus-time graph for a damped oscillator. 4/4/14 Physics 132 26 Damped Oscillations  A damped oscillator has position x = xmaxcos(ωt + φ0), where:  This slowly changing function xmax provides a border to the rapid oscillations, and is called the envelope.  The figure shows several oscillation envelopes, corresponding to different values of the damping constant b. 4/4/14 Physics 132 27 Energy in Damped Systems  Because of the drag force, the mechanical energy of a damped system is no longer conserved.  At any particular time we can compute the mechanical energy from:  Where the decay constant of this function is called the time constant τ, defined as:  The oscillator’s mechanical energy decays exponentially with time constant τ. 4/4/14 Physics 132 28 Driven Oscillations and Resonance  Consider an oscillating system that, when left to itself, oscillates at a natural frequency f0.  Suppose that this system is subjected to a periodic external force of driving frequency fext.  The amplitude of oscillations is generally not very high if fext differs much from f0.  As fext gets closer and closer to f0, the amplitude of the oscillation rises dramatically. 4/4/14 A singer or musical instrument can shatter a crystal goblet by matching the goblet’s natural oscillation Physics 132 frequency. 29 Driven Oscillations and Resonance The response curve shows the amplitude of a driven oscillator at frequencies near its natural frequency of 2.0 Hz. 4/4/14 Physics 132 30 Driven Oscillations and Resonance  The figure shows the same oscillator with three different values of the damping constant.  The resonance amplitude becomes higher and narrower as the damping constant decreases. 4/4/14 Physics 132 31 The graph shows how three oscillators respond as the frequency of a driving force is varied. If each oscillator is started and then left alone, which will oscillate for the longest time? A. B. C. D. The red oscillator. The blue oscillator. The green oscillator. They all oscillate for the same length of time. 66% e. m sa he e fo r t ee gr .. or . ill at or . n os c ill at os c Th ey a ll os c ill at Th e bl ue Th e Th e re d os c ill at or . 12% 13% 9% 1/23/13 Physics 132 • https://www.youtube.com/watch?v=xo x9BVSu7Ok 1/23/13 Physics 132 33