Physics 201: Lecture 28, Pg 1 Lecture 28 Goals Goals  Describe oscillatory motion  Use oscillatory graphs  Define the phase constant  Employ energy conservation

Physics 201: Lecture 28, Pg 2 Final Exam Details l Sunday, May 13th 10:05am-12:05pm in 125 Ag Hall l Format:  Closed book  Up to 4 8 ½ x11 sheets, written only  Approximately 50% from Chapters and 50% 1-12  Bring a calculator l Special needs/ conflicts: All requests for alternative test arrangements should be made by Thursday May10th (except for medical emergency)

Physics 201: Lecture 28, Pg 3 Periodic Motion Examples of periodic motion l Earth around the sun l Elastic ball bouncing up an down l Quartz crystal in your watch, computer clock, iPod clock, etc.

Physics 201: Lecture 28, Pg 4 Periodic Motion is everywhere Examples of periodic motion l Heart beat Taking your pulse, you count 70.0 heartbeats in 1 min. What is the period in seconds? Period is the time for one oscillation T = 60 sec/ 70.0 = 0.86 s l What is the frequency? (1 cycle/sec = 1 Hertz) f = 1 / T = 1.17 Hz

Physics 201: Lecture 28, Pg 5 Simple Harmonic Motion (SHM) l Period : The time it takes the block to complete one cycle. Denoted as T Measured in seconds l Frequency: The number of cycles complete per unit time. Denoted as f and f = 1 / T. SI units, measured in 1/seconds, or hertz (Hz). l If the period is doubled, the frequency is A. unchanged B. doubled C. halved

Physics 201: Lecture 28, Pg 6 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 may be linear ) e.g. Hooke’s law spring F = -k x A pendulum has the behavior only linear for small angles 3. Inertia 4. The drag forces are reasonably small

Physics 201: Lecture 28, Pg 7 Simple Harmonic Motion (SHM) l 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 l Hooke’s Law : F = -k x If k >> m  rapid oscillations large frequency If k low frequency

Physics 201: Lecture 28, Pg 8 Simple Harmonic Motion (SHM) l 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 A 0(≡X eq )

Physics 201: Lecture 28, Pg 9 A Spring Block oscillator (a) Spring under tension (b) Spring at equilibrium (c) Spring under compression A plot of displacement versus time includes both the amplitude and the period Amplitude: Maximum magnitude displacement, +A or -A Period: Repeat time, T

Physics 201: Lecture 28, Pg 10 Simple Harmonic Motion l For (x,t) graph below l 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 Position

Physics 201: Lecture 28, Pg 11 Simple Harmonic Motion l The period is T. l 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 201: Lecture 28, Pg 12 Simple Harmonic Motion l Now assume that the t coordinate of point K is s. l What is the period T, in seconds? l 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 201: Lecture 28, Pg 13 Simple Harmonic Motion l Now assume that the t coordinate of point K is s. l What is the period T, in seconds? T = s l How much time t does the block take to travel from the point of maximum displacement, A, to the opposite point of maximum displacement, -A ? time

Physics 201: Lecture 28, Pg 14 Simple Harmonic Motion l Now assume that the x coordinate of point R is 0.12 m. l What total distance d does the object cover during one period of oscillation? l What distance d does the object cover between the moments labeled K and N on the graph? time

Physics 201: Lecture 28, Pg 15 Simple Harmonic Motion l Now assume that the x coordinate of point R is 0.12 m. l What total distance d does the object cover during one period of oscillation? d = 0.48 m l What distance d does the object cover between the moments labeled K and N on the graph? time

Physics 201: Lecture 28, Pg 16 SHM Dynamics: Newton’s Laws still apply l At any given instant we know that F = ma must be true. l But in this case F = -k x k x m F = -k x a Position x(t) and acceleration a(t) are related and the acceleration is not constant We can show that is a possible solution

Physics 201: Lecture 28, Pg 17 SHM Dynamics: Newton’s Laws still apply l Velocity k x m F = -k x a l Displacement l Acceleration

Physics 201: Lecture 28, Pg 18 SHM Dynamics: Solution l Solution if and only if k x m F = -k x a l And…

Physics 201: Lecture 28, Pg 19 SHM So Far A solution is x = A cos(  t) where A = amplitude  = (angular) frequency l Question: If I double the amplitude will the period (a) decrease (b) stay the same (c) increase

Physics 201: Lecture 28, Pg 20 SHM Solution... Below is a drawing of A cos(  t ) where A is the amplitude of oscillation  T = 2  /  A A l Also a solution l So use Phase constant is 

Physics 201: Lecture 28, Pg 21 SHM dynamics k x m 0 Initial conditions determine phase constant  sin cos 

Physics 201: Lecture 28, Pg 22 SHM Dynamics l x(t), v(t) and a(t)  T = 2  /  A A tt x(t)v(t)a(t)U or K 0A0 -2A-2A All U  /2 0 -A-A 0All K  -A0 2A2A All U 3  /2 0 AA 0All K

Physics 201: Lecture 28, Pg 23 Mech. Energy of the Spring-Mass System Kinetic energy is always K = ½ mv 2 = ½ m(  A) 2 sin 2 (  t+  ) Potential energy of a spring is, U = ½ k x 2 = ½ k A 2 cos 2 (  t +  ) And  2 = k / m or k = m  2 U = ½ m  2 A 2 cos 2 (  t +  ) x(t) = A cos(  t +  ) v(t) = -  A sin(  t +  ) a(t) = -  2 A cos(  t +  )

Physics 201: Lecture 28, Pg 24 Mech. Energy of the Spring-Mass System And the mechanical energy is K + U =½ m  2 A 2 cos 2 (  t +  ) + ½ m  2 A 2 sin 2 (  t +  ) K + U = ½ m  2 A 2 [cos 2 (  t +  ) + sin 2 (  t +  )] K + U = ½ m  2 A 2 = ½ k A 2 which is constant    U~cos 2 K~sin 2 E = ½ kA 2

Physics 201: Lecture 28, Pg 25 SHM The most general solution is x(t) = A cos(  t +  ) where A = amplitude  = (angular) frequency  = phase constant l For SHM without friction,  The frequency does not depend on the amplitude !  The oscillation occurs around the equilibrium point where the force is zero!  Mechanical energy is a constant, it transfers between potential and kinetic

Physics 201: Lecture 28, Pg 26 Tuesday l All of chapter 15