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

$Force = Spring constant x displacement\,$

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

$f = ma = {d^2x \over dt^2}\,$

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

$\omega_0^2 = {s \over m}\,$

To solve the equation, we can assume

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