2. Physics 151: Lecture 30, Pg 1 Physics 151: Lecture 33 Today’s Agenda l Topics çPotential energy and SHM çResonance

3. Physics 151: Lecture 30, Pg 2 Energy in SHM l For both the spring and the pendulum, we can derive the SHM solution using energy conservation. l The total energy (K + U) of a system undergoing SMH will always be constant! l This is not surprising since there are only conservative forces present, hence energy is conserved. -AA0 s U U K E See text: 13.3

4. Physics 151: Lecture 30, Pg 3 SHM and quadratic potentials l SHM will occur whenever the potential is quadratic. l Generally, this will not be the case: l For example, the potential between H atoms in an H 2 molecule looks something like this: -AA0 x U U K E See text: Fig. 13.11 U x

5. Physics 151: Lecture 30, Pg 4 SHM and quadratic potentials... However, if we do a Taylor expansion of this function about the minimum, we find that for small displacements, the potential IS quadratic: U x U(x) = U(x 0 ) + U(x 0 ) (x- x 0 ) + U  (x 0 ) (x- x 0 ) 2 +.... U(x) = 0 (since x 0 is minimum of potential) x0x0 U x Define x = x - x 0 and U(x 0 ) = 0 Then U(x) = U  (x 0 ) x 2 See text: Fig. 13.11

6. Physics 151: Lecture 30, Pg 5 SHM and quadratic potentials... U x x0x0 U x U(x) = U  (x 0 ) x 2 Let k = U  (x 0 ) Then: U(x) = k x 2 SHM potential !! See text: Fig. 13.11

7. Physics 151: Lecture 30, Pg 6 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.

8. Physics 151: Lecture 30, Pg 7 What about Friction? We can guess at a new solution. With,

9. Physics 151: Lecture 30, Pg 8 What about Friction? What does this function look like? (You saw it in lab, it really works)

10. Physics 151: Lecture 30, Pg 9 What about Friction? There is a cuter way to write this function if you remember that exp(ix) = cos x + i sin x.

11. Physics 151: Lecture 30, Pg 10 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

12. Physics 151: Lecture 30, Pg 11 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, See text: 13.6 F = F 0 cos(  t)

13. Physics 151: Lecture 30, Pg 12 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, See text: 13.6

14. Physics 151: Lecture 30, Pg 13 Driven SHM with Resistance l So this is what we need to think about the amplitude of the oscillating motion, See text: 13.6 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,

15. Physics 151: Lecture 30, Pg 14 Driven SHM with Resistance l Now, consider what b does,   b small b middling b large See text: 13.6

16. Physics 151: Lecture 30, Pg 15 Lecture 33, 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)

17. Physics 151: Lecture 30, Pg 16 Lecture 33, Act 1 Solution l The frequency of a pendulum is l The driving frequency is l For each pendulum the natural frequency is  D   for pendulum B so it is in resonance. The answer is (B)

18. Physics 151: Lecture 30, Pg 17 Recap of today’s lecture l Chapter 13 çResonance