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

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

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

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

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

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

or,

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

Note that

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

To solve the equation, we can assume

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

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