Lecture7
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Lecture7

Course Number: PHY 101, Fall 2008

College/University: Syracuse

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PHY 101 Lecture #7: ElasticEnergy,SimpleHarmonicOscillators PHY 101 Lecture #7 Elastic Energy,<a href="/keyword/simple-harmonic-oscillator/" >simple <a href="/keyword/harmonic-oscillator/" >harmonic oscillator</a> </a> s 1 Springs usually resist stretching or squeezing Elastic Energy =kx2/2 PHY 101 Lecture#7 Elastic Energy,<a...

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PHY 101 Lecture #7: ElasticEnergy,SimpleHarmonicOscillators PHY 101 Lecture #7 Elastic Energy,<a href="/keyword/simple-harmonic-oscillator/" >simple <a href="/keyword/harmonic-oscillator/" >harmonic oscillator</a> </a> s 1 Springs usually resist stretching or squeezing Elastic Energy =kx2/2 PHY 101 Lecture#7 Elastic Energy,<a href="/keyword/simple-harmonic-oscillator/" >simple <a href="/keyword/harmonic-oscillator/" >harmonic oscillator</a> </a> s 2 Examples of springs for storage of elastic potential energy Hand-cranked radio Old-fashioned mechanical watch or clock Advantages: Compact Robust and reliable You can add energy with just your hand. Disadvantage: Limited storage capacity have to wind radio ~ every 15 minutes, wind watch daily PHY 101 Lecture #7 Elastic Energy,<a href="/keyword/simple-harmonic-oscillator/" >simple <a href="/keyword/harmonic-oscillator/" >harmonic oscillator</a> </a> s 3 Outline 1. <a href="/keyword/simple-harmonic-oscillator/" >simple <a href="/keyword/harmonic-oscillator/" >harmonic oscillator</a> </a> s Characteristics of motion Energy analysis Examples of oscillators 2. Friction vs. the conservation of energy? PHY 101 Lecture #7 Elastic Energy,<a href="/keyword/simple-harmonic-oscillator/" >simple <a href="/keyword/harmonic-oscillator/" >harmonic oscillator</a> </a> s 4 1. <a href="/keyword/simple-harmonic-oscillator/" >simple <a href="/keyword/harmonic-oscillator/" >harmonic oscillator</a> </a> s Earlier, we observed motion of a mass due to force of gravity. Today, observe motion of a mass due to spring force. Motion is distinctive: <a href="/keyword/simple-harmonic/" >simple harmonic</a> motion. Properties: periodic (i.e. repetitive) sinusoidal similar in form at different amplitudes PHY 101 Lecture #7 Elastic Energy,<a href="/keyword/simple-harmonic-oscillator/" >simple <a href="/keyword/harmonic-oscillator/" >harmonic oscillator</a> </a> s 5 Motion of mass on spring If you remember trig, you can see that this motion is sinusoidal: x = A sin (2ft ). PHY 101 Lecture #7 Elastic Energy,<a href="/keyword/simple-harmonic-oscillator/" >simple <a href="/keyword/harmonic-oscillator/" >harmonic oscillator</a> </a> s 6 Concepts for periodic motion Periodic motion: the same motion repeats over and over. Amplitude A: the max displacement in a cycle. Period T: the time taken by one complete cycle. Frequency f: the number of cycles per second, so f = 1/T. Unit of frequency is the cycle/second, called the hertz, or Hz.) (Sometimes it is handy to use the angular frequency = 2 f.) (Unit of angular frequency is the radian/second.) PHY 101 Lecture #7 Elastic Energy,<a href="/keyword/simple-harmonic-oscillator/" >simple <a href="/keyword/harmonic-oscillator/" >harmonic oscillator</a> </a> s 7 Motion of mass on spring Measured period T is about 1.85 s, or frequency f = 0.54 Hz. Textbook gives: f = 1 2 k . m Plug in k = 8.5 N/m, m = 0.7 kg. Checks. PHY 101 Lecture #7 Elastic Energy,<a href="/keyword/simple-harmonic-oscillator/" >simple <a href="/keyword/harmonic-oscillator/" >harmonic oscillator</a> </a> s 8 Same kind of motion at any amplitude Amplitude of motion depends on how &quot;hard&quot; you start it. Period (and thus frequency) stay the same. Motion still a sine wave, just bigger or smaller. PHY 101 Lecture#7 Elastic Energy,<a href="/keyword/simple-harmonic-oscillator/" >simple <a href="/keyword/harmonic-oscillator/" >harmonic oscillator</a> </a> s 9 Velocity of <a href="/keyword/simple-harmonic-oscillator/" >simple <a href="/keyword/harmonic-oscillator/" >harmonic oscillator</a> </a> Velocity is sinusoidal, too. Maximum speed where stretch = 0. Speed = 0 at maximum of motion. Blue = position Red = velocity PHY 101 Lecture#7 Elastic Energy,<a href="/keyword/simple-harmonic-oscillator/" >simple <a href="/keyword/harmonic-oscillator/" >harmonic oscillator</a> </a> s 10 Energy of <a href="/keyword/simple-harmonic-oscillator/" >simple <a href="/keyword/harmonic-oscillator/" >harmonic oscillator</a> </a> Two important forms of energy involved in <a href="/keyword/simple-harmonic/" >simple harmonic</a> motion: kinetic energy of mass K=mv2/2 elastic potential energy of spring Uelast=kx2/2 Examine each throughout periodic motion. Then examine total energy E = K + Uelast. PHY 101 Lecture#7 Elastic Energy,<a href="/keyword/simple-harmonic-oscillator/" >simple <a href="/keyword/harmonic-oscillator/" >harmonic oscillator</a> </a> s 11 Kinetic energy of <a href="/keyword/simple-harmonic-oscillator/" >simple <a href="/keyword/harmonic-oscillator/" >harmonic oscillator</a> </a> K (red curve) is sinusoidal (at twice the freq. of the motion.) K is max where speed is max (stretch is zero), zero where speed is zero (stretch is max.) PHY 101 Lecture #7 Elastic Energy,<a href="/keyword/simple-harmonic-oscillator/" >simple <a href="/keyword/harmonic-oscillator/" >harmonic oscillator</a> </a> s 12 Elastic potential energy of <a href="/keyword/simple-harmonic-oscillator/" >simple <a href="/keyword/harmonic-oscillator/" >harmonic oscillator</a> </a> Elastic energy (magenta) is sinusoidal (at twice the freq. of the motion.) Elastic energy is max where stretch is max, zero where stretch is zero. PHY 101 Lecture #7 Elastic Energy,<a href="/keyword/simple-harmonic-oscillator/" >simple <a href="/keyword/harmonic-oscillator/" >harmonic oscillator</a> </a> s 13 Total energy of <a href="/keyword/simple-harmonic-oscillator/" >simple <a href="/keyword/harmonic-oscillator/" >harmonic oscillator</a> </a> Total energy (black curve), sum of K and Uelast, is constant (i.e., conserved.) Etotal = Uelast grows when K shrinks, and shrinks when K grows. PHY 101 Lecture#7 Elastic Energy,<a href="/keyword/simple-harmonic-oscillator/" >simple <a href="/keyword/harmonic-oscillator/" >harmonic oscillator</a> </a> s 14 1 2 kA . 2 Why does a <a href="/keyword/simple-harmonic-oscillator/" >simple <a href="/keyword/harmonic-oscillator/" >harmonic oscillator</a> </a> oscillate? Start motion with zero stretch, maximum speed. Oscillator has K, but no Uelast. As spring compresses, Uelast builds up. This must happen at expense of K. So motion slows down. Motion continues upward until it stops when all K has been converted to Uelast. Spring's downward force starts to move mass downward. Compression is reduced, so Uelast decreases. K increases. Fastest motion at zero stretch. Same pattern occurs downward, with spring stretching until all K has become Uelast, then motion starts upward. Repeats again.... PHY 101 Lecture #7 Elastic Energy,<a href="/keyword/simple-harmonic-oscillator/" >simple <a href="/keyword/harmonic-oscillator/" >harmonic oscillator</a> </a> s 15 Examples of oscillators Pendulum Gravity provides the spring. Metronome Tuning forks Each tine is part &quot;mass&quot; and part &quot;spring&quot;. Stringed musical instrument String is the mass; springiness from tension. Wind instrument Air in tube is the &quot;mass&quot; and the &quot;spring&quot;. Balance wheel in mechanical clock/watch PHY 101 Lecture #7 Elastic Energy,<a href="/keyword/simple-harmonic-oscillator/" >simple <a href="/keyword/harmonic-oscillator/" >harmonic oscillator</a> </a> s 16 Why are <a href="/keyword/simple-harmonic-oscillator/" >simple <a href="/keyword/harmonic-oscillator/" >harmonic oscillator</a> </a> s so useful? Fixed period, independent of amplitude, makes them good &quot;frequency reference&quot;. For musical instrument, know you'll get the same note whether loud or soft. For clock, keeps good time whether spring is wound up or almost wound down. <a href="/keyword/simple-harmonic-oscillator/" >simple <a href="/keyword/harmonic-oscillator/" >harmonic oscillator</a> </a> s are everywhere! Almost anything that vibrates is a SHO. You can make electrical versions, too. Now more common than mechanical oscillators. PHY 101 Lecture#7 Elastic Energy,<a href="/keyword/simple-harmonic-oscillator/" >simple <a href="/keyword/harmonic-oscillator/" >harmonic oscillator</a> </a> s 17 2. Is energy really conserved? Observe the motion of a <a href="/keyword/simple-harmonic-oscillator/" >simple <a href="/keyword/harmonic-oscillator/" >harmonic oscillator</a> </a> for a long time (many periods of oscillation.) Does it continue to oscillate with same amplitude? Where does the energy go? Is energy really conserved? PHY 101 Lecture#7 Elast...
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