Lecture 19Slide 2Periodic Motion is everywhereSlide 4A special kind of periodic oscillator: Harmonic oscillator What do all “harmonic oscillators” have in common?Simple Harmonic Motion (SHM)Slide 7Slide 8Slide 9Simple Harmonic MotionSlide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17SHM Dynamics: Newton’s Laws still applySHM Solution...Combining sin and cosine solutionsEnergy of the Spring-Mass SystemSlide 22Slide 23Slide 24SHM So FarThe “Simple” PendulumThe shaker cartSlide 28Slide 29Slide 30Physics 207: Lecture 19, Pg 1Lecture 19Goals:Goals:•Chapter 14Chapter 14 Interrelate the physics and mathematics of oscillations. Draw and interpret oscillatory graphs. Learn the concepts of phase and phase constant. Understand and use energy conservation in oscillatory systems. Understand the basic ideas of damping and resonance.Phase Contrast MicroscopeEpithelial cell in brightfield (BF) using a 40x lens (NA 0.75) (left) and with phase contrast using a DL Plan Achromat 40x (NA 0.65) (right). A green interference filter is used for both images.Physics 207: Lecture 19, Pg 2Lecture 19•AssignmentAssignment HW8, Due Wednesday, Apr. 7th Thursday: Read through Chapter 15.4Physics 207: Lecture 19, Pg 3Periodic Motion is everywhereExamples of periodic motionEarth around the sunElastic ball bouncing up an downQuartz crystal in your watch, computer clock, iPod clock, etc.Physics 207: Lecture 19, Pg 4Periodic Motion is everywhereExamples of periodic motionHeart beat In taking your pulse, you count 70.0 heartbeats in 1 min. What is the period, in seconds, of your heart's oscillations? Period is the time for one oscillation T= 60 sec/ 70.0 = 0.86 sWhat is the frequency? f = 1 / T = 1.17 HzPhysics 207: Lecture 19, Pg 5A special kind of periodic oscillator: Harmonic oscillatorWhat do all “harmonic oscillators” have in common?1. A position of equilibrium2. A restoring force, (which may be linear ) [Hooke’s law spring F = -k x (In a pendulum the behavior only linear for small angles: sin θ where θ = s / L) ] In this limit we have: F = -ks with k = mg/L)3. Inertia4. The drag forces are reasonably smallPhysics 207: Lecture 19, Pg 6Simple Harmonic Motion (SHM)In Simple Harmonic Motion the restoring force on the mass is linear, that is, exactly proportional to the displacement of the mass from rest positionHooke’s Law : F = -k xIf k >> m rapid oscillations <=> large frequencyIf k << m slow oscillations <=> low frequencyPhysics 207: Lecture 19, Pg 7Simple Harmonic Motion (SHM)We know that if we stretch a spring with a mass on the end and let it go the mass will, if there is no friction, ….do something1. Pull block to the right until x = A2. After the block is released from x = A, it willA: remain at restB: move to the left until it reaches equilibrium and stop thereC: move to the left until it reaches x = -A and stop thereD: move to the left until it reaches x = -A and then begin to move to the rightkmkmkm-AA0(≡Xeq)Physics 207: Lecture 19, Pg 8Simple Harmonic Motion (SHM)The time it takes the block to complete one cycle is called the period. Usually, the period is denoted T and is measured in seconds.The frequency, denoted f, is the number of cycles that are completed per unit of time: f = 1 / T. In SI units, f is measured in inverse seconds, or hertz (Hz).If the period is doubled, the frequency is A. unchanged B. doubled C. halvedPhysics 207: Lecture 19, Pg 9Simple Harmonic Motion (SHM) An oscillating object takes 0.10 s to complete one cycle; that is, its period is 0.10 s. What is its frequency f ?Express your answer in hertz. f = 1/ T = 10 HzPhysics 207: Lecture 19, Pg 10Simple Harmonic MotionNote in the (x,t) graph that the vertical axis represents the x coordinate of the oscillating object, and the horizontal axis represents time.Which points on the x axis are located a displacement A from the equilibrium position ?A. R onlyB. Q onlyC. both R and QtimePositionPhysics 207: Lecture 19, Pg 11Simple Harmonic MotionSuppose that the period is T.Which of the following points on the t axis are separated by the time interval T?A. K and LB. K and MC. K and PD. L and NE. M and PtimePhysics 207: Lecture 19, Pg 12Simple Harmonic MotionNow assume that the t coordinate of point K is 0.0050 s.What is the period T , in seconds?How much time t does the block take to travel from the point of maximum displacement to the opposite point of maximum displacement? timePhysics 207: Lecture 19, Pg 13Simple Harmonic MotionNow assume that the t coordinate of point K is 0.0050 s.What is the period T , in seconds?T = 0.02 sHow much time t does the block take to travel from the point of maximum displacement to the opposite point of maximum displacement? timePhysics 207: Lecture 19, Pg 14Simple Harmonic MotionNow assume that the t coordinate of point K is 0.0050 s.What is the period T , in seconds?T = 0.020 sHow much time t does the block take to travel from the point of maximum displacement to the opposite point of maximum displacement? t = 0.010 stimePhysics 207: Lecture 19, Pg 15Simple Harmonic MotionNow assume that the x coordinate of point R is 0.12 m.What total distance d does the object cover during one period of oscillation? What distance d does the object cover between the moments labeled K and N on the graph? timePhysics 207: Lecture 19, Pg 16Simple Harmonic MotionNow assume that the x coordinate of point R is 0.12 m.What total distance d does the object cover during one period of oscillation?d = 0.48 mWhat distance d does the object cover between the moments labeled K and N on the graph? timePhysics 207: Lecture 19, Pg 17Simple Harmonic MotionNow assume that the x coordinate of point R is 0.12 m.What total distance d does the object cover during one period of oscillation?d = 0.48 mWhat distance d does the object cover between the moments labeled K and N on the graph?d = 0.36 mtimePhysics 207: Lecture 19, Pg 18SHM Dynamics: Newton’s Laws still applyAt any given instant we know that F = ma must be true.But in this case F = -k x and ma = So: -k x = ma =kxmFF = -k x aad xdtkmx22a differential equation for x(t) !22dtxdm22dtxdm“Simple approach”, guess a solution and see if it works!Physics 207: Lecture 19, Pg 19SHM Solution...Try either cos ( t ) or sin ( t ) Below is a drawing of A
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