In our differential equation, we have approximated by
. When is the magnitude of the error caused by
this approximation less than, say, 10% of the magnitude of the
actual function?
The actual function is , so 10% of that is
. The absolute error between
and
is
. Thus, the absolute error is less than 10% of
the magnitude of the actual function when
For what values of is
close to
? To help you, a graph shows
in blue and
in green.
What if you wanted a 5% error? Estimate the valid range of theta
for this case.
We have addressed the issue of how close is to
, but not the issue of
how close the solution of the linearized pendulum equation is to
the solution of the nonlinear pendulum equation. We will return
to this more difficult and important issue later in the module.