Physics 151: Lecture 33 Today’s Agenda

1. Physics 151: Lecture 33 Today’s Agenda • Topics • Potential energy and SHM • Resonance

2. U K E U s -A 0 A See text: 13.3 Energy in SHM • For both the spring and the pendulum, we can derive the SHM solution using energy conservation. • The total energy (K + U) of a system undergoing SMH will always be constant! • This is not surprising since there are only conservative forces present, hence energy is conserved.

3. U U K E U x x -A 0 A See text: Fig. 13.11 SHM and quadratic potentials • SHM will occur whenever the potential is quadratic. • Generally, this will not be the case: • For example, the potential betweenH atoms in an H2 molecule lookssomething like this:

4. U(x) = U(x0 ) + U(x0 ) (x- x0 ) + U (x0 ) (x- x0 )2+.... U U x0 x Definex = x - x0 andU(x0 ) = 0 Then U(x) = U (x0 )x2 x  See text: Fig. 13.11 SHM and quadratic potentials... However, if we do a Taylor expansion of this function about the minimum, we find that for smalldisplacements, the potential IS quadratic: U(x) = 0 (since x0 is minimum of potential)

5. See text: Fig. 13.11 SHM and quadratic potentials... U(x) = U (x0)x2 Letk = U (x0) Then: U(x) = kx2 U U x0 x x  SHM potential !!

6. What about Friction? • Friction causes the oscillations to get smaller over time • This is known as DAMPING. • As a model, we assume that the force due to friction is proportional to the velocity.

7. What about Friction? We can guess at a new solution. With,

8. What about Friction? What does this function look like? (You saw it in lab, it really works)

9. What about Friction? There is a cuter way to write this function if you remember that exp(ix) = cos x + i sin x .

10. Damped Simple Harmonic Motion • Frequency is now a complex number! What gives? • Real part is the new (reduced) angular frequency • Imaginary part is exponential decay constant underdamped overdamped critically damped

11. See text: 13.6 Driven SHM with Resistance • To replace the energy lost to friction, we can drive the motion with a periodic force. (Examples soon). • Adding this to our equation from last time gives, F = F0 cos(wt)

12. See text: 13.6 Driven SHM with Resistance • So we have the equation, • As before we use the same general form of solution, • Now we plug this into the above equation, do the derivatives, and we find that the solution works as long as,

13. Something more surprising happens if you drive the pendulum at exactly the frequency it wants to go, See text: 13.6 Driven SHM with Resistance • So this is what we need to think about the amplitude of the oscillating motion, • Note, that A gets bigger if Fo does, and gets smaller if b or m gets bigger. No surprise there. • Then at least one of the terms in the denominator vanishes and the amplitude gets real big. This is known as resonance.

14. b small b middling b large See text: 13.6 Driven SHM with Resistance • Now, consider what b does, w = w0

15. Lecture 33, Act 1Resonant Motion • Consider the following set of pendula all attached to the same string A D B If I start bob D swinging which of the others will have the largest swing amplitude ? (A) (B) (C) C

16. Lecture 33, Act 1Solution • The frequency of a pendulum is • The driving frequency is • For each pendulum the natural frequency is • wD = w0 for pendulum B so it is in resonance. The answer is (B)

17. Recap of today’s lecture • Chapter 13 • Resonance