Consider a mass that is connected to a spring on a frictionless horizontal surface. To understand the oscillatory motion of the system, apply DID TASC . This gives:

ΣF = ma   →   -kx = ma

The acceleration is the second time derivative of the position:

d2x dt2

= -

k m

x

The solutions to this equation of motion are periodic functions of time and have the generic form:

x(t) = A cos ω t   or   x(t) = B sin ωt

Here A is the amplitude of the oscillation, and ω is the angular frequency.

Taking two time derivatives of x(t) gives:

v(t)

=

dx dt

= -Aω sin ωt

a(t)

=

d2x dt2

=

dv dt

= -Aω2 cos ωt   →   a = -ω2 x

Comparing with the original equation of motion   a = - k x/m, the angular frequency is:   ω² = k/m.

The frequency ω is related to the period T of the oscillation by:   T = 2 π/ ω

Summary:

Simple harmonic motion (SHM) is described by x(t) = A cos ωt (or x(t) = A sin ωt).

The governing equation of SHM is:   a = - ω² x.