LO1 Curtis Koo

1. 1. At an angle of 0o, a pendulum is released from a horizontal position and swings
down 90o to its vertical position.
Are pendulums an example of SHM? Why or why not?
Solution:
When the pendulum swings, it accelerates
towards the point of suspension and towards
its equilibrium point (where acceleration is
zero and velocity is maximum). We say that
the motion of a pendulum is an example of
simple harmonic motion because of the
linear, tangential acceleration.
m g sinθ in this diagram shows the restoring
force of the pendulum.
In SHM, restoring force should be
proportional to the displacement from the
equilibrium point which forms a length
along the chord.
In the diagram, s = Lθ and using the
equation for SHM, we can equate the
restoring force to –ks, therefore we get
m g sinθ = - k Lθ
k therefore equals to mg sinθ / Lθ
Using some simple calculus, we see that sin
theta is used to approximate x for small
angles.

2. Because of the small angle approximation, we are able to neglect the sinθ / θ and
therefore k = mg/ L and we are able to derive this equation:
However, in the case here, where there is a 90o angle (not a small angle), this
valuation of SMH falls apart since:
sin (pi/2)/(pi/2) = 0.6366;
Which does not follow the small angle approximation which would have given an
answer of 1.
As a result, for a simple pendulum, this approximation is only accurate for small
angles because of the expression for angular acceleration.