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, $$f = ma = m{d^2x \over dt^2}\,$$ 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 $$x(t)=A e^{\lambda t} \,$$

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.