Physics Fall 2014 Lecture Welcome back to Physics 215 Oscillations Simple harmonic motion

Physics Fall 2014 Lecture 12-22

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

Physics Fall 2014 Lecture 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...

Physics Fall 2014 Lecture 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)

Physics Fall 2014 Lecture Simple Harmonic Oscillator Differential equation for x (t): Solution:

Physics Fall 2014 Lecture 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

Physics Fall 2014 Lecture Simple Harmonic Oscillator DEMO stronger spring (larger k)  faster oscillations (larger f) larger mass  slower oscillations

Physics Fall 2014 Lecture Simple Harmonic Oscillator Total Energy E =

Physics Fall 2014 Lecture Simple Harmonic Oscillator -- Summary If F =  k x then

Physics Fall 2014 Lecture 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)

Physics Fall 2014 Lecture (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

Physics Fall 2014 Lecture 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.

Physics Fall 2014 Lecture 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.

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

Physics Fall 2014 Lecture 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?

Physics Fall 2014 Lecture Torsion Pendulum (Angular Simple Harmonic Oscillator)  =    Torsion constant DEMO Solution:

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

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

Physics Fall 2014 Lecture Resonance Small (b) damping Large (b) damping A dd

Physics Fall 2014 Lecture Reading assignment Gravity Chapter 13 in textbook