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

${\displaystyle {\rm {Force={\rm {Spring~constant\times {\rm {displacement\,}}}}}}}$

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

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

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

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

or,

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

Note that

${\displaystyle \omega _{0}^{2}={s \over m}\,}$

To solve the equation, we can assume

${\displaystyle x(t)=Ae^{\lambda t}\,}$

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