Simple harmonic motion: showing the relationships between position, velocity and acceleration.

Simple harmonic motion can be derived from Newton's second law and Hooke's law. It has an equation of motion described by this acceleration function:[br][br][math]a = \frac{\textrm{d}^2 x}{\textrm{d} t^2} = -\omega ^2 x[/math][br][br]The force acting on a particle (and hence, the acceleration) is proportional to the distance of a particle from its equilibrium and is directed towards the equilibrium point. It is a restoring force always trying put the particle back in equilibrium.[br][br][list][br][*]The constant [math]\omega [/math] is the angular frequency ([math]\textrm{rad s}^{-1}[/math]). It is related to the mass (constant) and stiffness of the spring (constant). Increasing [math]\omega [/math] will produce quicker oscillations.[br][*]The constant [math]\phi[/math] is the phase angle; it describes where in the cycle the oscillations begin. [math]\phi = 0[/math] means the particle starts at rest from maximum displacement. Different values of [math]\phi [/math] mean a non zero initial velocity closer to the equilibrium point.[br][*]The amplitude is the maximum height of the oscillations measured from equilibrium.[br][*]The magnitude and direction of the position, velocity and acceleration vectors are show for comparison.[br][/list]