The Not-so-simple Pendulum 2

This problem takes the equation for simple harmonic motion to the next stage of complexity. The solution to the equation is provided giving students an opportunity to explore the interpretation of an equation/solution and to see where calculus might start to go at university.

There are three obvious ways in which this problem could be used:

First way: As a discussion point into modelling assumptions, work with the solution as given. This would work in any context in which students are beginning to develop an understanding of differentiation equations. This particular equation is interesting because it offers a reasonable level of complexity whilst still allowing for a solution involving only simple functions.

Start by discussing friction. How does this affect motion? Why is the $v$ term a sensible candidate for modelling this? How does the equation relate to $F=ma$?

Next discuss the expected qualitative form of a damped pendulum motion. Then proceed to look at the form of the solution to see if it compares with this. At one level, the solution might clearly be seen both to decay and oscillate, but more detail will be needed to see if the decay occurs at a sensible rate. Students will need to choose some sensible values for the parameters in order to do this. They are unlikely to have any sense of the appropriate scale for $\lambda$, so might need to try out a range of values.

Students could try to plot the solution (either numerically, or as a curve sketching exercise) and show that this would seem to represent a damped pendulum motion.

Second way: The solution to this problem is a relatively extended piece of mathematics. Can students work their way through this? This gives an excellent revision into concepts from calculus such as chain rule, product rule and differentiation of standard functions, and gives a good flavour of the complexities of 1st year undergraduate mathematics.

Third way: Students could try to differentiate the solution twice and substitute back into the equation to show that it works. This would be best tackled just prior to coming to university: suggest that students interested in a mathematics or engineering degree work through the problem to gain insight into what is to come and also to give their differentiation a solid workout.

Intuitively, how does friction affect motion? Does the $v$ term accomplish this to some level?

How does the equation relate to $F=ma$?

As a rough sketch without any calculation, what would you expect the trajectory over time of the exact, real pendulum motion be over time? Compare this to a plot of the solution.

What would be realistic values of the parameters? Make some well-reasoned estimations if you have no intuitive sense for the scale of the parameters.

This equation is not the final part in the story of modelling the simple pendulum. Students might try to devise an accurate experiment to determine its accuracy, or to suggest alternative forms of the equation with other friction or drag terms in.

Try The Not-So-Simple Pendulum 1 first.