ComplexNumbersSHO

ComplexNumbersSHO - previous index next From a Circling...

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previous index next From a Circling Complex Number to the Simple Harmonic Oscillator Michael Fowler Describing Real Circling Motion in a Complex Way We’ve seen that any complex number can be written in the form i zr e θ = , where r is the distance from the origin, and is the angle between a line from the origin to z and the x -axis. This means that if we have a set of numbers all with the same r , but different ’s, such as , ii re re α β , etc., these are just different points on the circle with radius r centered at the origin in the complex plane. Now think about a complex number that moves as time goes on: ( ) . it z tA e ω = At time t , z ( t ) is at a point on the circle of radius A at angle t to the x -axis. That is, z ( t ) is going around the circle at a steady angular velocity . We can also write this: () cos sin z e A t i A t == + and see that the point z = x + iy is at coordinates ( ) ( ) ,c o s , s i n . x yA t A t = x y O ( ) zt t sin A t A cos At The angular velocity is , the actual velocity in the complex plane is dz ( t )/ dt .
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This note was uploaded on 02/09/2012 for the course PHYSICS 152 taught by Professor Michaelfowler during the Fall '07 term at UVA.

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ComplexNumbersSHO - previous index next From a Circling...

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