1. Physics 1501: Lecture 27, Pg 1 Physics 1501: Lecture 27 Today’s Agenda l Homework #9 (due Friday Nov. 4) l Midterm 2: Nov. 16 l Katzenstein Lecture: Nobel Laureate Gerhard t’Hooft çFriday at 4:00 in P-36 … l Topics çSHM çDamped oscillations çResonance ç1-D traveling waves

2. Physics 1501: Lecture 27, Pg 2 l Spring-mass system l Pendula çGeneral physical pendulum »Simple pendulum çTorsion pendulum Simple Harmonic Oscillator  d Mg z-axis R x CM  =  0 cos(  t +  ) k x m F F = -kxa I wire   x(t) = Acos(  t +  ) where

3. Physics 1501: Lecture 27, Pg 3 What about Friction? l Friction causes the oscillations to get smaller over time l This is known as DAMPING. l As a model, we assume that the force due to friction is proportional to the velocity.

4. Physics 1501: Lecture 27, Pg 4 What about Friction? We can guess at a new solution. With,

5. Physics 1501: Lecture 27, Pg 5 What about Friction? What does this function look like? (You saw it in lab, it really works)

6. Physics 1501: Lecture 27, Pg 6 What about Friction? There is a cuter way to write this function if you remember that exp(ix) = cos x + i sin x.

7. Physics 1501: Lecture 27, Pg 7 Damped Simple Harmonic Motion l Frequency is now a complex number! What gives? çReal part is the new (reduced) angular frequency çImaginary part is exponential decay constant underdamped critically damped overdamped Active Figure

8. Physics 1501: Lecture 27, Pg 8 Driven SHM with Resistance l To replace the energy lost to friction, we can drive the motion with a periodic force. (Examples soon). l Adding this to our equation from last time gives, F = F 0 cos(  t)

9. Physics 1501: Lecture 27, Pg 9 Driven SHM with Resistance l So we have the equation, l As before we use the same general form of solution, l Now we plug this into the above equation, do the derivatives, and we find that the solution works as long as,

10. Physics 1501: Lecture 27, Pg 10 Driven SHM with Resistance l So this is what we need to think about, I.e. the amplitude of the oscillating motion, l Note, that A gets bigger if F o does, and gets smaller if b or m gets bigger. No surprise there. l Then at least one of the terms in the denominator vanishes and the amplitude gets real big. This is known as resonance. l Something more surprising happens if you drive the pendulum at exactly the frequency it wants to go,

11. Physics 1501: Lecture 27, Pg 11 Driven SHM with Resistance l Now, consider what b does,   b small b middling b large  

12. Physics 1501: Lecture 27, Pg 12 Dramatic example of resonance l In 1940, turbulent winds set up a torsional vibration in the Tacoma Narrow Bridge 

13. Physics 1501: Lecture 27, Pg 13 Dramatic example of resonance  l when it reached the natural frequency

14. Physics 1501: Lecture 27, Pg 14 Dramatic example of resonance  l it collapsed ! Other example: London Millenium Bridge

15. Physics 1501: Lecture 27, Pg 15 Lecture 27, Act 1 Resonant Motion l Consider the following set of pendula all attached to the same string D A B C If I start bob D swinging which of the others will have the largest swing amplitude ? (A)(B)(C)

16. Physics 1501: Lecture 27, Pg 16 Chap. 13: Waves What is a wave ? l A definition of a wave: çA wave is a traveling disturbance that transports energy but not matter. l Examples: çSound waves (air moves back & forth) çStadium waves (people move up & down) çWater waves (water moves up & down) çLight waves (what moves ??) Animation

17. Physics 1501: Lecture 27, Pg 17 Types of Waves l Transverse: The medium oscillates perpendicular to the direction the wave is moving. çWater (more or less) çString waves l Longitudinal: The medium oscillates in the same direction as the wave is moving çSound çSlinky

18. Physics 1501: Lecture 27, Pg 18 Wave Properties Wavelength Wavelength: The distance between identical points on the wave. Amplitude A l Amplitude: The maximum displacement A of a point on the wave. A Animation

19. Physics 1501: Lecture 27, Pg 19 Wave Properties... l Period: The time T for a point on the wave to undergo one complete oscillation. Speed: The wave moves one wavelength in one period T so its speed is v =  / T. Animation

20. Physics 1501: Lecture 27, Pg 20 Wave Properties... We will show that the speed of a wave is a constant that depends only on the medium, not on amplitude, wavelength or period and T are related ! v = / T = v T or = 2  v /  (since  T = 2  /   or  v / f (since T = 1/ f ) l Recall f = cycles/sec or revolutions/sec  rad/sec = 2  f

21. Physics 1501: Lecture 27, Pg 21 Lecture 27, Act 2 Wave Motion l The speed of sound in air is a bit over 300 m/s, and the speed of light in air is about 300,000,000 m/s. l Suppose we make a sound wave and a light wave that both have a wavelength of 3 meters. çWhat is the ratio of the frequency of the light wave to that of the sound wave ? (a) About 1,000,000 (b) About.000,001 (c) About 1000

22. Physics 1501: Lecture 27, Pg 22 Wave Forms continuous waves l So far we have examined “continuous waves” that go on forever in each direction ! v v l We can also have “pulses” caused by a brief disturbance of the medium: v l And “pulse trains” which are somewhere in between.

23. Physics 1501: Lecture 27, Pg 23 Mathematical Description l Suppose we have some function y = f(x): x y l f(x-a) is just the same shape moved a distance a to the right: x y x=a 0 0 l Let a=vt Then f(x-vt) will describe the same shape moving to the right with speed v. x y x=vt 0 v

24. Physics 1501: Lecture 27, Pg 24 Math... Consider a wave that is harmonic in x and has a wavelength of. If the amplitude is maximum at x=0 this has the functional form: y x A l Now, if this is moving to the right with speed v it will be described by: y x v

25. Physics 1501: Lecture 27, Pg 25 Math... l By usingfrom before, and by defining l So we see that a simple harmonic wave moving with speed v in the x direction is described by the equation: we can write this as: (what about moving in the -x direction ?)

26. Physics 1501: Lecture 27, Pg 26 Math Summary l The formula describes a harmonic wave of amplitude A moving in the +x direction. y x A Each point on the wave oscillates in the y direction with simple harmonic motion of angular frequency . l The wavelength of the wave is l The speed of the wave is l The quantity k is often called “wave number”. Movie (twave)