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Though the spring is the most common example of simple harmonic motion, a pendulum can be approximated by simple harmonic motion, and the torsional oscillator obeys simple harmonic motion. Both of these examples will be examined in depth in Applications of Simple Harmonic Motion.
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 (F = ma), 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:
[pic]
m [pic] [pic] [pic] = - kx
or [pic]
[pic]
(Note that: This is equation of Simple harmonic)
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Let us interpret this equation. The second derivative of a function of 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:
[pic]
Where “a” and “b” are constants. Differentiating this equation, we see that
[pic]
And
[pic]
Plugging this into our original differential equation, we see that:
[pic]
It is clear that, if b 2 = [pic] , then the equation is satisfied. Thus the equation governing simple harmonic oscillation is:
Simple
[pic]
Similarly the equation of motion of damped harmonic oscillator is
• Thus equation of motion of damped harmonic oscillator is
[pic]
where,r=(γ/2m) and ω2=k/m
• Solution of above equation is of the form
x=Ae-rtcos(ω't+φ)
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