1. Physics 215 -- Fall 2014 Lecture 12-21 Welcome back to Physics 215 Oscillations Simple harmonic motion

2. Physics 215 -- Fall 2014 Lecture 12-22

3. Physics 215 -- Fall 2014 Lecture 12-23 Current homework assignment HW10: –Knight Textbook Ch.14: 32, 52, 56, 74, 76, 80 –Due Wednesday, Nov. 19 th in recitation

4. Physics 215 -- Fall 2014 Lecture 12-24 Oscillations Restoring force leads to oscillations about stable equilibrium point Consider a mass on a spring, or a pendulum Oscillatory phenomena also in many other physical systems...

5. Physics 215 -- Fall 2014 Lecture 12-25 Simple Harmonic Oscillator x 0 F=0 F FF F x =  k x Newton’s 2 nd Law for the block: Spring constant Differential equation for x (t)

6. Physics 215 -- Fall 2014 Lecture 12-26 Simple Harmonic Oscillator Differential equation for x (t): Solution:

7. Physics 215 -- Fall 2014 Lecture 12-27 Simple Harmonic Oscillator f – frequency Number of oscillations per unit time T – Period Time taken by one full oscillation Units: A - m T - s f - 1/s = Hz (Hertz)  - rad/s amplitude angular frequency initial phase

8. Physics 215 -- Fall 2014 Lecture 12-28 Simple Harmonic Oscillator DEMO stronger spring (larger k)  faster oscillations (larger f) larger mass  slower oscillations

9. Physics 215 -- Fall 2014 Lecture 12-29 Simple Harmonic Oscillator Total Energy E =

10. Physics 215 -- Fall 2014 Lecture 12-210 Simple Harmonic Oscillator -- Summary If F =  k x then

11. Physics 215 -- Fall 2014 Lecture 12-211 Importance of Simple Harmonic Oscillations For all systems near stable equilibrium –F net ~ - x where x is a measure of small deviations from the equilibrium –All systems exhibit harmonic oscillations near the stable equilibria for small deviations Any oscillation can be represented as superposition (sum) of simple harmonic oscillations (via Fourier transformation) Many non-mechanical systems exhibit harmonic oscillations (e.g., electronics)

12. Physics 215 -- Fall 2014 Lecture 12-212 (Gravitational) Pendulum Simple Pendulum – Point-like Object DEMO   0x F net = mg sin  L m For small  F net is in – x direction: F x =  mg/L x mg T F net “Pointlike” – size of the object small compared to L

13. Physics 215 -- Fall 2014 Lecture 12-213 Two pendula are created with the same length string. One pendulum has a bowling ball attached to the end, while the other has a billiard ball attached. The natural frequency of the billiard ball pendulum is: 1. greater 2. smaller 3. the same as the natural frequency of the bowling ball pendulum.

14. Physics 215 -- Fall 2014 Lecture 12-214 The bowling ball and billiard ball pendula from the previous slide are now adjusted so that the length of the string on the billiard ball pendulum is shorter than that on the bowling ball pendulum. The natural frequency of the billiard ball pendulum is: 1. greater 2. smaller 3. the same as the natural frequency of the bowling ball pendulum.

15. Physics 215 -- Fall 2014 Lecture 12-215 (Gravitational) Pendulum Physical Pendulum – Extended Object DEMO  net = d mg sin   For small  : d m, I   mg T sin  ≈   net = d mg 

16. Physics 215 -- Fall 2014 Lecture 12-216 A pendulum consists of a uniform disk with radius 10cm and mass 500g attached to a uniform rod with length 500mm and mass 270g. What is the period of its oscillations?

17. Physics 215 -- Fall 2014 Lecture 12-217 Torsion Pendulum (Angular Simple Harmonic Oscillator)  =    Torsion constant DEMO Solution:

18. Physics 215 -- Fall 2014 Lecture 12-218 Damped Harmonic Oscillator For example air drag for small speeds: F drag = - b v Damping constant Differential equation for x(t) Solution:

19. Physics 215 -- Fall 2014 Lecture 12-219 Forced Harmonic Oscillator Driving force Solution yields amplitude of response: Damping force Natural frequency Differential equation for x(t):

20. Physics 215 -- Fall 2014 Lecture 12-220 Resonance Small (b) damping Large (b) damping A dd

21. Physics 215 -- Fall 2014 Lecture 12-221 Reading assignment Gravity Chapter 13 in textbook