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# The Not-so-simple Pendulum 2

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Age 16 to 18

Challenge Level

- Problem
- Getting Started
- Student Solutions
- Teachers' Resources

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.