Lecture 19 Goals Chapter 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 Microscope Epithelial 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 1 Periodic Motion is everywhere Examples of periodic motion Earth around the sun Elastic ball bouncing up an down Quartz crystal in your watch computer clock iPod clock etc Physics 207 Lecture 19 Pg 2 Periodic Motion is everywhere Examples of periodic motion Heart 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 s What is the frequency f 1 T 1 17 Hz Physics 207 Lecture 19 Pg 3 A special kind of periodic oscillator Harmonic oscillator What do all harmonic oscillators have in common 1 A position of equilibrium 2 A restoring force which must be linear Hooke s law spring F k x In a pendulum the behavior only linear for small angles sin where s L In this limit we have F ks with k mg L 3 Inertia 4 The drag forces are reasonably small Physics 207 Lecture 19 Pg 4 Simple 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 position Hooke s Law F kx If k m rapid oscillations large frequency If k m slow oscillations low frequency Physics 207 Lecture 19 Pg 5 Simple 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 something 1 Pull block to the right until x A 2 After the block is released from x A it will A remain at rest B move to the left until it reaches equilibrium and stop there C move to the left until it reaches x A and stop there D move to the left until it reaches x A and then begin to move to the right k m k m k m A 0 Xeq A Physics 207 Lecture 19 Pg 6 Simple Harmonic Motion SHM We know that if we stretch a spring with a mass on the end and let it go the mass will 1 Pull block to the right until x A 2 After the block is released from x A it will k A remain at rest m B move to the left until it reaches equilibrium and stop there k m C move to the left until it reaches x A and stop there k D move to the left until it reaches m x A and then begin to move to A 0 Xeq A the right This oscillation is called Simple Harmonic Motion Physics 207 Lecture 19 Pg 7 Simple 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 halved Physics 207 Lecture 19 Pg 8 Simple 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 halved Physics 207 Lecture 19 Pg 9 Simple 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 Hz Physics 207 Lecture 19 Pg 10 Simple Harmonic Motion Note in the x t graph that the vertical axis represents the x Position 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 only B Q only C both R and Q time Physics 207 Lecture 19 Pg 11 Simple Harmonic Motion Suppose that the period is T Which of the following points on the t axis are separated by the time interval T A K and L B K and M C K and P D L and N E M and P time Physics 207 Lecture 19 Pg 12 Simple Harmonic Motion Now 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 time Physics 207 Lecture 19 Pg 13 Simple Harmonic Motion Now assume that the t coordinate of point K is 0 0050 s What is the period T in seconds T 0 020 s How 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 s time Physics 207 Lecture 19 Pg 14 Simple Harmonic Motion Now 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 m What distance d does the object cover between the moments labeled K and N on the graph d 0 36 m time Physics 207 Lecture 19 Pg 15 SHM Dynamics Newton s Laws still apply At any given instant we know that F ma must be true But in this case F k x d 2x m 2 and ma dt d 2x So k x ma m 2 dt d 2x k x 2 m dt F k x k a m x a differential equation for x t Simple approach guess a solution and see if it works Physics 207 Lecture 19 Pg 16 SHM Solution Try either cos t or sin t Below is a drawing of A cos t where A amplitude of oscillation T 2 A A with k m and 2 f 2 T Both sin and cosine work so need to include both Physics 207 Lecture 19 Pg 17 Combining sin and cosine solutions B cos t C sin t A cos t A cos t cos sin t sin A cos cos t A sin sin t Notice that B A cos C A …
View Full Document
Unlocking...