Challenge Level

This problem takes one of the simplest differential equations
of mechanics -- the simple harmonic approximation to pendulum
motion -- and gives students the opportunity to probe the
assumptions underlying its derivation. This gives both practice
into calculating the impact of numerical changes in equations along
with some insights into mathematical modelling and non-linear
differential equations, which will be of interest and use to
students wishing to study a STEM course at university.

The problem can be considered during a mechanics course, but
is also well suited to students coming to the end of their school
career who wish to prepare themselves for their degree
course.

The ideas in this problem are fascinating and students should
be encouraged to solve the problem reflectively, thinking about the
implication of any of their numerical calculations.

Throughout the problem, this key focus should be stressed:
when does the assumption for SHM
give rise to good approximations? Students used to a
prescriptive approach to differential equations might need to be
encouraged to develop their own criteria for a 'good
approximation'. There is no need to be vague about this, and the
better modellers will understand this.

Are you clear as to what you are trying to do?

Have you related the answer back to the physical
situations?

The problem naturally continues with Not-so-simple Pendulum
2 in which the focus is on the solution of more difficult
differential equations. You could also read about modelling assumptions
in mechanics.

You could ignore the part about deriving the equation and
simply work from the equation directly.