Physics 5B Lecture 7 January 25 2012 Chapter 14 Pendulum P d l Damped Harmonic Oscillator Resonance R First Midterm Exam Next Monday January 30 Covers C the h first f three h homework h k assignments through Section 14 5 Note that the 3rd assignment is due Friday evening Solutions will be posted afterwards A practice exam is posted on eCommons An ordinary calculator is allowed but no notes computers tablets telephones Do not store notes in the calculator memory DRC in Thimann Labs 397 Simple p Plane Pendulum Energy gy E 12 mv 2 mg 1 cos E 12 m 2 2 12 mg 2 cos 1 12 2 for small angles for small angles looks like a S H O l cos http phet colorado edu en simulation pendulum lab g Simple Plane Pendulum 2 1 mg 1 E m 2 2 2 E 1 mv 2 1 kx 2 2 2 Pendulum Mass on spring mg For pendulum k so mg m g But the pendulum is only approximately a simple harmonic oscillator and the approximation is good only when the amplitude is small e g a few degrees Quartz pendulums used in the Gulf gravimeter 1929 Accurate to better than one part in 10 million A pendulum is hanging vertically from the ceiling of an l I i i ll the h elevator l i at rest and d the h period i d off the h elevator Initially is pendulum is T Then the elevator accelerates upward During the acceleration the pperiod of the ppendulum becomes A greater than T B equal to T C less than T T 2 g What if the cable is cut and the elevator falls freely Then what will the period be You feel heavier in an elevator acceleratingg upward p That is due to the floor pushing upward with greater force but it is equivalent to standing on a planet with a greater gravitational acceleration Problem 14 14 48 A physical pendulum consists of a tiny bob of mass M and a uniform if cord d off mass m and d length l h l l a Find a formula for the frequency using the small angle approximation b How does this compare with the formula for a simple pendulum ignoring the mass of the cord l M m Damped Harmonic Motion m d 2x dt 2 dx kx b dt The solution depends on how the damping constant compares with the spring constant and mass Let x t Ae t cos t x t B C t e t 2 2 k m b 2m underdamped critically damped x t A1e t A2 e t e t where Simple model for the friction overdamped 2 2 Underdamped Oscillator http phy hk wiki englishhtm Damped htm Damped p Oscillator Underdamped Critically damped Overdamped Example plots for two different initial conditions Resonance Demonstration Important Resonance is one of the most important concepts in all of physics Examples are ubiquitous Forced Damped p Oscillator Small damping h h Q high Mechanical example move the wall to which the spring is attached back and forth sinusoidally Eventually the mass will oscillate with the driving frequency not the natural oscillator frequency 0 Large damping low Q 0 f i ti friction http www walter fendt de ph14e resonance htm k m Forced Damped p Oscillator Important points After start up transients have died out the response is at the same frequency f as the h drive di The response amplitude is small far from resonance and large at resonance For high Q small damping the response can get huge at resonance when the drive operates at the natural frequency of the oscillator The phase difference between drive and response is zero far below resonance drive and oscillator move together 90 degrees at resonance e g the oscillator is at its maximum amplitude when the drive passes through its equilibrium position 180 degrees d far f above b resonance drive d and d oscillator ll move opposite each other The width of the resonant curve amplitude vs frequency is wide d for f large l d damping low l Q Q narrow for small damping high Q