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Math 5900 – Summer 2011 Lecture 1: Simple Harmonic Oscillations Gernot Laicher University of Utah - Department of Physics & Astronomy
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Newton's Second Law
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Acceleration
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Velocity
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Hook’s Law
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The system can be described with Newton’s second law as follows: Differential equation (“Second Order”: Contains second derivative; “Linear”: The function and its derivatives appear as powers of 1)
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Solution to this differential equation: Note: Alternatively, we could also have written the general solution in a different but equivalent form:
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Reinserting solution into DE:
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The amplitude A is determined by the initial conditions of the system (at “t=0”) and the resonance frequency : The phase angle is similarly determined by these initial conditions and the resonance frequency as follows: f: frequency of oscillation T: period of oscillation A: amplitude of oscillation
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Net restoring force directly proportional to displacement DE of that same form Simple harmonic (sinusoidal) oscillations In our example: : fixed by k and m A and imposed by the initial conditions Note: Changing is equivalent to shifting the time when t=0
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Spring Constantk=0.6N/m Massm=2kg AmplitudeA=1.2m Phase 0 degree s 0.54772 3s Frequencyf= 0.08717 3Hz Period 11.4714 7s Example:
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Energy of the oscillating mass (assuming no losses due to friction) Elastic Potential Energy: Total Energy: Kinetic Energy:
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