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

$Force=Springconstantxdisplacement$

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

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

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

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