A Level Maths Notes: M3 – Basics of Simple Harmonic Motion
Suppose the motion is
projected onto the
–
axis. Then
(1)
We can differentiate to
obtain the velocity:
(2)
and again to get the
acceleration:
(3)
We can write (3) as
so
This is often taken as the
basic equation of simple harmonic motion, and means that the the
acceleration is proportional to the acceleration, but directed
towards the zero point on the–
axis.
We can use the identity
to
obtain further equations.
From (1)
and
from (2)
Substituting these into the
identity gives
Similarly for projection
onto the
–
axis:
(4)
We can differentiate to
obtain the velocity:
(5)
and again to get the
acceleration:
(5)
We can write (5) as
so
We can use the identityto
obtain a further equation.
From (4)
and
from (5)
Substituting these into the
identity gives
If the frequency is
then
so
we can also write
and
with
corresponding equations for
