1. For a simple pendulum, the
pendulum swing back and
forth in a period motion…
So…

2. First, let’s look at simple harmonic
motion…
For a simple harmonic motion, the acceleration
equation is expressed as
a = - ω² x
knowing that ω is a constant value, the
acceleration of mass is directly proportional to
the displacement x.

3. we can rewrite the tangential acceleration as
at = - gsin(s/L)
In a simple pendulum, the
tangential acceleration is
expressed as
- gsinθ = at
according to the formula s = Lθ,
where s is the arc length, L is the
length of the string.

4. Let’s compare the acceleration equation in
simple harmonic motion and the acceleration
equation in simple pendulum…
at = - gsin(s/L) and a = - ω² x
Notice that the acceleration in simple pendulum
is not proportional to displacement, instead, it is
proportional to sin(s/L).
Therefore, pendulum motion is NOT simple
harmonic motion.