High School Physics/Simple Oscillation

Simple Oscillation

For a simple oscillator consisting of a mass m to one end of a spring with a spring constant s, the restoring force, f, can be expressed by the equation

$\mathrm{F}\mathrm{o}\mathrm{r}\mathrm{c}\mathrm{e}\mathrm{=}\mathrm{S}\mathrm{p}\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}\mathrm{\times }\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{p}\mathrm{l}\mathrm{a}\mathrm{c}\mathrm{e}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{t}$

where displacement is the displacement of the mass from its rest position. Substituting the expression for f into the linear momentum equation,

$f=ma=m\frac{d^{2}x}{d{t}^2}$

where a is the acceleration of the mass, we can get

$m\frac{d^{2}x}{d{t}^2}=-sx$

or,

$\frac{d^{2}x}{d{t}^2}+\frac{s}{m}x=0$

Note that

${\omega }_0^2=\frac{s}{m}$

To solve the equation, we can assume

$x(t)=A{e}^{\lambda t}$

Template:BookCat The general solution for this type of 'simple harmonic motion' is $x=A\sin (wt+\varphi )$. Here, $\varphi$ (the angle expressed in radians) is known as the phase of the simple harmonic motion.