- #1
ItsTheSebbe
- 10
- 1
Homework Statement
When I was in high school I was thaught that the period of a simple harmonic oscillation (mass on spring, ball on pendulum, etc) was equal to ##T=2\pi \sqrt \frac m k## though they have never explained to me why. That's what I wanted to find out.
So for example, let's take a mass ##m## on a spring with spring constant ##k##, give it a little nudge and it will start oscillating in a wave-like manner (let's assume no energy is lost due to friction).
PS: This is the first time I'm trying to use LaTeX (first post in general), so if there's anything wrong with the formatting, please let me know :)!
Homework Equations
##T=\frac {2\pi} {\omega}##
##F_{spring}=-kx##
##F_{net}=ma##
The Attempt at a Solution
##F_{net}=F_{spring}##
##ma=-kx##
##m \frac {d^2x} {dt^2} +kx=0##
##\frac {d^2x} {dt^2} +\frac k m x=0##
##\ddot x(t) + \frac k m x(t)=0##
##\text{take} \ \ x=e^{Rt}##
##R^2e^{Rt} +\frac k m e^{Rt}=0##
##R^2+\frac k m =0##
##R^2=-\frac k m##
##R=\pm i\sqrt \frac k m ##
##\text{hence, } x(t)=C_1e^{i\sqrt \frac k m t}+C_2e^{-i\sqrt \frac k m t}##
Now, here I get stuck. I know I need to eventually get ##x(t)=Acos(\omega t+\phi)##
I don't really understand how to get there; probably by using Euler's formula.
From here on out I can proof the equation, namely:
##\ddot x=-A\omega^2\cos(\omega t+\phi)=-x\omega^2##
##\text{plugging into EoM gives:}##
##-x\omega^2+\frac k m x=0##
##\omega^2=\frac k m##
##\omega=\sqrt \frac k m##
##T=\frac {2\pi} {\omega}=2\pi \sqrt \frac m k##