Harmonic Oscillator Mass-Spring Oscillator Resonance The Pendulum Physics 109 Experiment Number 12
Outline Simple harmonic motion The vertical mass-spring system Driven oscillations and resonance The pendulum Problems
Oscillatory Motion or Simple Define some terms: Harmonic Motion 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
We write the equation as: m d 2 x kx dt 2 This is a Differential Equation Time We assume that the solution is of the form x=acos( t)
Phase Phase is the angular relationship between two oscillators operating at the same frequency. Time The red curve is 180 degrees out of phase with the black curve
Checking the solution: dx dt A sin t d 2 x dt 2 A 2 cos t m A 2 cos t k Acos 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
Bad Shocks Auto, 1240 kg mass, shock absorber k = 15 kn/m, M/4 = 310 kg Period = T = 2p (m/k) 1/2 = 0.903s Frequency = 1/T =1.11 Hz Why are the shocks bad?
Example a swaying skyscraper
The Tacoma Narrows Bridge, 1940
Tacoma Narrows Movie http://www.pbs.org/wgbh/nova/bridge/tacoma3.html http://www.youtube.com/watch?v=3mclp9qmcgs http://www.5min.com/video/the-original-tacoma-narrows- Bridge-Collapse-of-1940-119995718 http://www.youtube.com/watch?v=j-zczjxsxnw
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.
Procedure 1. Determine the k for the spring in the usual way. 2. Attach a mass to the spring 3. Pull down the mass, and release 4. Measure the period of oscillation 5. Verify that the angular frequency is (k/m) 1/2.
Driven Oscillations and Resonance ma kx b v F 0 cos d t Restoring Damping Driving x Acos d t 2 A F 0 m d 2 2 0 b 2 2 d m 2 If damping is small, large amplitudes can be built up..
The Pendulum (also part of expt) Tension does not exert torque, but gravity does. mgl sin
Pendulum formulae Rotational analog Torque = -mgl sin ( ) Ia So, for small amplitudes, = -mgl. 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.
Class Problems General Oscillation Questions
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. 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 D. One fourth cycle of harmonic motion
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 c3:c113 1.5 1 0.5 0-0.5-1 -1.5 P 0 20 40 60 80 100 120 Series1 C. Positive velocity and zero acceleration b3:b113 D. Negative velocity and positive acceleration E. Zero velocity and zero acceleration.
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. B. Stays the same. C. Decreases.
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. 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 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
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. An object at rest must be in equilibrium.
An object can oscillate around: A. Any equilibrium point B. Any stable equilibrium point 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
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. One period later D. None of the above.