Simple harmonic motion is a type of oscillatory motion where the restoring force is directly proportional to the displacement and acts in the direction opposite to that of the displacement (see Hooke's law). This becomes the following differential equation:
which results in the following solution:
where A is the amplitude, ω is the angular frequency, equal to 2πf, and φ is the phase. ω is equal to
The total energy of the system at any time is
Derivation of formulaEdit
Hooke's law can be rewritten as a second-order differential equation.
This equation will have the characteristic equation
The solution to the differential equation is
By using Euler's formula this can be written in the form
which, by using trigonometric identities, can be written as
where and .