A Level Physics Notes: Waves and Oscillations – Properties of Simple Harmonic Motion
The general equation of simple harmonic motion is
and
the general solution to this equation is
The
amplitude of the oscillations is
The
constants A and B are arbitrary. In particular, they do not depend on
the period of oscillations. This may be surprising, but it is a
universal feature of simple harmonic motion that the period and the
amplitude are independent of each other. On the other hand, the period does depend on the physical constants
of the apparatus, while the amplitude is determined by the initial conditions - how the motion starts.
For instance, the simple pendulum obeys the equation
and
then
and
the
length of the pendulum, is a property of the pendulum.
A mass m on a spring with spring constant k obeys the equation
and
then
In theory almost any oscillation may be approximated by simple harmonic motion if the oscillation in small enough.
For small oscillations about the point A, the motion may be simple harmonic because the potential for a small interval at A is approximately symmetric and A is a minimum.