from the equilibrium position), is the angular frequency, and is the initial phase. Using the techniques of calculus, the velocity and acceleration as a function of time can be found: Speed: Maximum speed: (at equilibrium point) Maximum acceleration: (at extreme points) By definition, if a mass is under SHM its acceleration is directly proportional to displacement. where Since , and, since where is the time period, These equations demonstrate that the simple harmonic motion is isochronous (the period and frequency are independent of the amplitude and the initial phase of the motion). Energy Substituting with , the kinetic energy of the system at time is and the potential energy is In the absence of friction and other energy loss, the total mechanical energy has a constant value Examples The following physical systems are some examples of simple harmonic oscillator. Mass on a spring A mass attached to a spring of spring constant exhibits simple harmonic motion in closed space. The equation for describing the period shows the period of oscillation is independent of the amplitude, though in practice the amplitude should be small. The above equation is also valid in the case when an additional constant force is being applied on the mass, i.e. the additional constant force cannot change the period of oscillation. Uniform circular motion Simple harmonic motion can be considered the one-dimensional projection of uniform circular motion. If an object moves with angular speed around a circle of radius centered at the origin of the -plane, then its motion along each coordinate is simple harmonic motion with amplitude and angular frequency . Oscillatory Motion It is the motion of a body when it moves to and fro about a definite point. This type of motion is also called Oscillatory motion or vibratory motion. The time period is able to be calculated by where l is the distance from rotation to centre of mass of object undergoing SHM and g being gravitational field constant. This is analogous to the mass-spring system. Mass of a simple pendulum In the small-angle approximation, the motion of a simple pendulum is approximated by simple harmonic motion. The period of a mass attached to a pendulum of length with gravitational acceleration is given by This shows that the period of oscillation is independent of the amplitude and mass of the pendulum but not of the acceleration due to gravity, , therefore a pendulum