Oscillations - Part 1

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Oscillations is the study of Periodic motion. In periodic motion there is an equilibrium position and the object in motion swings to and fro from the highest point of amplitude A to the lowest point -A. This to and fro is caused by "Springiness", bouncing the object from one side to the other. It is expressed with k. [math] F = -kx[/math] [Force exerted by Spring] [math] F_{ext} = +kx[/math] External Force on Spring Just note, that in this case the force on the spring is not constant, so no constant acceleration eq’s. For dealing with Oscillations, it's time for some new vocabulary and symbols. Displacement: Distance from equilibrium point (x) Amplitude: The greatest distance from the equilibrium point (A) Cycle: One complete to and fro motion around a single point period T: The time required to complete one cycle. T = 1/f (remember, capital T is time required to complete one cycle, little t is the current time in the motion) Frequency: The number of complete cycles per second. (in Hertz Hz) F = 1/T Hertz = 1 cycle per second Simple Harmonic Motion “When the net restoring force is directly proportional to the negative of displacement” a system exhibiting this is a simple harmonic oscillator [math] A[/math] = Amplitude [math] phi [/math] = phase angle, how long after (or before) t = 0 peak at x = A is reached [math] omega [/math] = angular frequency (rad/sec) To find the eq for a simple harmonic oscillator use F = ma, but there is a trick, acceleration is a derivative [math]displaystyle a = frac {d^2x}{dt^2}[/math] so Newton’s second law, Net force equals mass times acceleration is written as: [math] displaystyle m frac{d^2x}{dt^2} = -kx[/math] which can be rearranged to the “Equation of Motion” [math]displaystyle frac{d^2x}{dt^2} + frac{k}{m}x = 0[/math] The Equation of Motion is a differential Equation. The solution to x in the Equation of Motion is [math] x = A cos(omega t+ phi)[/math] and is solving for omega, we have: [math] omega ^x = frac{k}{m}[/math]