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The Not-so-simple Pendulum 1

Age 16 to 18 Challenge Level:

Why do this problem?

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.

Possible approach

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.

Key questions

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

Have you related the answer back to the physical situations?

Possible extension

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.

Possible support

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