Equation for displacement in damped harmonic motion.

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The discussion focuses on the equation for displacement in damped harmonic motion, specifically x=Cexp(-b/2mt)·exp(±√(b²/2m-k/m)t). The constant C is determined by the initial conditions of the system, typically using the initial displacement x₀ at time t=0. To find C, one substitutes the known values into the equation and solves for it as a constant of integration. This approach allows for the accurate modeling of the system's behavior over time. Understanding how to calculate C is crucial for analyzing damped oscillations effectively.
Craptola
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This is not really a homework problem but rather a question about an equation for displacement in damped harmonic oscillations that I've come across during revision for midterms. In my notes and in various textbooks the equation is given as x=C\mathrm{exp}(-\frac{b}{2m}t)\cdot\mathrm{exp}(\pm \sqrt{\frac{b^{2}}{2m}-\frac{k}{m}}t)

I've been told that C is a constant depending on the initial conditions of the system, but I'm not sure how to go about determining the value of this constant.
Any help on this matter would be greatly appreciated.
 
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You use a value for x that you know for some value of t. Usually ##t=0## and ##x=x_0## - you know, how far you pulled the pendulum back before you let it go?
You put the numbers into the equation and solve for C. It's a constant of integration.
 
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