Simple Harmonic Motion

Simple Harmonic Motion - x plus the function itself (times...

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Simple Harmonic Motion >From our concept of a simple harmonic oscillator we can derive rules for the motion of such a system. We start with our basic force formula, F = - kx . Using Newton's Second Law , we can substitute for force in terms of acceleration: ma = - kx Here we have a direct relation between position and acceleration. For you calculus types, the above equation is a differential equation, and can be solved quite easily. Note : The following derivation is not important for a non- calculus based course, but allows us to fully describe the motion of a simple harmonic oscillator. Deriving the Equation for Simple Harmonic Motion Rearranging our equation in terms of derivatives, we see that: m = - kx or + x = 0 Let us interpret this equation. The second derivative of a function of
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Unformatted text preview: x plus the function itself (times a constant) is equal to zero. Thus the second derivative of our function must have the same form as the function itself. What readily comes to mind is the sine and cosine function. Let us come up with a trial solution to our differential equation, and see if it works. As a tentative solution, we write: x = a cos( bt ) where a and b are constants. Differentiating this equation, we see that = - ab sin( bt ) and = - ab 2 cos( bt ) Plugging this into our original differential equation, we see that: - ab 2 cos( bt ) + a cos( bt ) = 0 It is clear that, if b 2 = , then the equation is satisfied. Thus the equation governing simple harmonic oscillation is: simple x = a cos t...
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