Experimenting with a Pendulum: Damping & Plotting Graphs

AI Thread Summary
The discussion revolves around an experiment on pendulum damping, focusing on deriving a second-order differential equation and its solutions. The equation mx" + (k/m)x' + nx = 0 is identified as a linear homogeneous differential equation with constant coefficients. The characteristic equation derived from this is mr² + (k/m)r + n = 0, which leads to complex solutions indicating oscillatory behavior. To plot a curve, initial conditions are necessary to determine constants D1 and D2 in the solution form eat(D1cos(bt) + D2sin(bt)). The conversation emphasizes the need for clarity on how to extract the damping coefficient and plot the results based on measured amplitudes.
amppatel
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I am doing an experiment on the pendulum, and i wanted to do something on the damping, i understand a little as i have done some reading on it, the picture attached show the working i have done, i have reached a second order DE and i know little about this. So my question is, can you plot a graph? and how can you find out the damping co efficent, i have some results i have measured the amplitude every other half oscilation.
 

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Okay, you have mx"+ (k/m)x'+ nx= 0. That's a linear homogeneous differential equation with constant coefficients, about the simplest possible equation. If you "try" a solution of the form x= ert, you arrive at the characteristic equation:
mr2+ (k/m)r+ n= 0. That is a quadratic equation and can be solved by the quadratic equation, giving r= a+bi and a- bi which means that the solutions to the differential equation are of the form C1e(a+ bi)t+ C2e(a-bi)t which can in turn be written eat(D1cos(bt)+ D2sin(bt)). You must have some initial conditions to determine D1 and D2.
 
ok so i get that, abit, but does that allow me to plot a curve? or can i solve the equation for k?
 
also eat(D1cos(bt)+ D2sin(bt)) what is that equal to? zero? Thanks for all your help
 
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