Harmonic Oscillator. Outline. Oscillatory Motion or Simple Harmonic Motion. Oscillatory Motion or Simple Harmonic Motion

Transcription

1 Harmonic Oscillator Mass-Spring Oscillator Resonance The Pendulum Physics 109, Class Period 13 Experiment Number 11 in the Physics 121 Lab Manual (page 65) Outline Simple harmonic motion The vertical mass-spring system Driven oscillations and resonance The pendulum Problems Assignment Oscillatory Motion or Simple Harmonic Motion Define some terms: Frequency, f, Period, T f = 1/T One cycle per second = One Hertz Simple harmonic motion results when an object is subject to a restoring force that is proportional to its displacement from equilibrium F = -kx the minus sign indicates a restoring force Oscillatory Motion or Simple Harmonic Motion F = - kx can also be written, ma = - kx Time 1

2 We write the equation as: m d 2 x kx dt 2 This is a Differential Equation We assume that the solution is of the form x=acos( t) Time Checking the solution: dx dt A sin t d 2 x dt 2 A 2 cos t m A 2 cos t k A cos t So the solution holds if m 2 k, k m Where is the angular frequency for simple harmonic motion. (the units are radians / second) The frequency, f, is /2 and the period, T, is 1/f Example a swaying skyscraper A vertical mass spring system This is one part of the experiment. When the spring balances gravity: mg kx 1 0 When stretched a bit more: F spring k x 1 x Net force F mg k x 1 x Gravity changes equilibrium position but does not change frequency. 2

3 Driven Oscillations and Resonance ma kx b v F 0 cos d t The Tacoma Narrows Bridge, 1940 Restoring Damping Driving x Acos d t 2 A F 0 m d b 2 2 d m 2 If damping is small, large amplitudes can be built up.. Tacoma Narrows Movie The Pendulum (also part of expt) Tension does not exert torque, but gravity does. mgl sin 3

4 Pendulum formulae Rotational analog Ia Torque = -mgl sin ( ) So, for small amplitudes, = -mgl. Problems from Diagnostic The direction of acceleration of block at position Q is best represented by which arrow? AC And the frequency is: mgl I mgl ml 2 g l Note that for a simple pendulum, the frequency (and period) are independent of the mass. P D Q C B R Problems from Diagnostic Assuming air resistance is negligible, which is true? A. Acceleration of ball was greatest just before point Q (still in contact with spring) B. Acceleration is decreasing from Q to R C. Acceleration is zero at R. D. Acceleration is the same for all points Q to R. C. Gravity R Q Based on Homework Class Problems General Oscillation Questions P 4

5 Weighing an Astronaut An astronaut is strapped to a spring in order to be weighed. What do we measure? A. The amount that the spring is stretched. B. The spring constant. C. The period of oscillation. C. D. None of the above. For two identical mass-spring systems, displaced the same amount, how much later should one be released for the phase difference to be 90 degrees? A. One half period later. B. One fourth period later. C. C. One period later D. None of the above. A mass slides along a frictionless surface and strikes a spring with velocity, v. How long is the mass in contact with the spring? A. One cycle of harmonic motion. B. Two cycles of harmonic motion C. One half cycle of harmonic motion C. D. One fourth cycle of harmonic motion Oscillation Questions An object is in equilibrium when net force and net torque = 0. Which statement is true for an object in an inertial frame of reference? A. Any object in equilibrium is at rest. B. An object in equilibrium need not be at rest. C. C. An object at rest must be in equilibrium. 5

6 An object can oscillate around: A. Any equilibrium point B. Any stable equilibrium point C. C. Certain or specific stable equilibrium points D. Any point. Which of the following is necessary to make an object oscillate? A. A stable equilibrium B. Little or no friction C. A disturbance D. All of the above C. A mass attached to a spring oscillates back and forth as indicated in the position vs. time plot. At point P, the mass has: A. Positive velocity and positive acceleration B. Positive velocity and negative acceleration C. C. Positive velocity and zero acceleration D. Negative velocity and positive acceleration E. Zero velocity and zero acceleration. c3:c P b3:b113 Series1 An object hangs motionless from a spring. When the object is pulled down, the sum of elastic potential energy of the spring and the gravitational potential energy: A. Increases. C. B. Stays the same. C. Decreases. 6

7 A person swings on a swing. Person sits still and swing oscillates at its natural frequency. Now, another person comes and sits, and there are now two people on the swing. The natural frequency is: A. Greater B. The same C. C. Smaller A person swings when person sits still, the swing oscillates at its natural frequency. If, instead, the person stands, the natural frequency is: A. Greater C. B. The same C. Smaller Two identical mass-spring systems are displaced the same amounts from equilibrium and then released at different times. Of the amplitudes, frequencies, periods and phase constants of the resulting motions, which is different? A. Amplitude B. Frequencies C. Periods D. Phase constants C. Assignment Read Chapter 11, page 267 ff. in the text Do problems 11-21,27,39,45 Read about the experiment on the Ballistic Pendulum, Experiment 7, p. 51 This will be our last experiment! Review of the Mechanics portion of the course will be done next Tuesday. No Lab or classes Wed. due to Thanksgiving. 7