Motion of a pendulum with air resistance

Question

I am trying to model the motion of a pendulum with air resistance. I have resolved perpendicular to the direction of motion to get this equation where $m$, $g$, $p$, $C_D$ and $A$ are constants: $$mg\sin⁡(θ)-\frac{1}{2} pv^2 C A=ma$$

This can be expressed as the following differential equation $$mg \sin⁡(θ) - \frac{1}{2} p\left(\frac{dθ}{dt}\right)^2 C =m\left(\frac{d^2 θ}{dt^2}\right)$$

How this equation would be solved?

Answer 1

Here you are using Reynolds law formula for drag. If you use Stokes law, and consider small amplitudes, you can simplify greatly your formula. See

http://nrich.maths.org/6478

http://nrich.maths.org/6478/solution

http://nrich.maths.org/content/id/6478/Paul-not%20so%20simple%20pendulum%202.pdf

http://nrich.maths.org/content/id/6478/Ben-Not%20so%20simple%20pendulum%202.pdf

Another interesting paper is

The pendulum - Rich physics from a simple system

by Robert A. Nelson and M. G. Olsson

Am. J. Phys., Vol. 54, No. 2, February 1986