Solving the Simple Harmonic Oscillator

Physics 5 BWinter 2009 Solving the Simple Harmonic Oscillator1. The Harmonic Oscillator solution: displacement as a function of timeWe wish to solve the equation of motion for the Simple Harmonic Oscillator :d2xdt2= kmx ,(1)wherekis the spring constant andmis the mass of the oscillating body that isattached to the spring. We impose the following initial conditions on the 0, the initial displacement is denoted byx0and the corresponding velocityis denoted byv0. That is,x(t= 0) x0,anddxdt(t= 0) =v0.(2)These initial conditions then uniquely specify the method we shall employ for Solving this differential equation is called themethod of inspired guessing. In class, we argued that the motion of the oscillatingbody was periodic. Since the sine and cosine functions are periodic, we proposethe following solution for the displacementxas a function of the timet:x(t) =x0cos t+bsin t ,(3)whereband are to be determined. If we sett= 0, we findx(t= 0) =x0asrequired, so one of the two initial conditions is automatically satisfied.

Solving the Simple Harmonic Oscillator 1. The harmonic oscillator solution: displacement as a function of time We wish to solve the equation of motion for the simple harmonic oscillator: d2x dt2 = − k m x, (1) where k is the spring constant and m is the mass of the oscillating body that is attached to the spring.

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  Oscillators, Harmonics, Harmonic oscillator, The harmonic oscillator

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