Topic 2: The pendulum

PHY321F — cp 2005 25
The problem of determining the position as a function of time, or the period,
is now a matter of finding the value of some elliptic integrals. As there is
no ‘calculator key’ for these, we either have to interpolate from tables or
evaluate the integrals numerically. These are both computational problems.
We can, of course, also solve the differential equation on the computer.
We can make the pendulum model more realistic by including damping. The
linearly damped harmonic oscillator and its solution are well-known:
d 2 θ
dt + 2 Ω2 θ + β dθ
dt = 0
where β = b/(mL) with b a damping parameter and m the bob mass. Then
the solution is
θ = θ m e − β 2 t cos(Ω t + φ)
Of course we should really do this for the mathematical pendulum — then
there is no analytical solution. In addition, the damping term is inappropriate
for a pendulum bob moving at a typical speed in air. The damping force is
better described by
F d = 1 2 CρAv|v|
(where ρ is the density of air, C is a drag coefficient, A an effective crosssectional
area and v is the speed L ˙θ of the bob) rather than F d = bv. C can
depend on the speed as well. This gives a drag term in the above equation
C dθ
ρAL
2m dt
dθ
∣ dt ∣ = γ dθ
dθ
dt ∣ dt ∣
Let us see how to solve this problem computationally.
2.2 Numerical methods
2.1 Solving the differential equation
While there exist techniques for integrating second-order differential equations,
it is more convenient to exploit first-order integrators. We start by
writing the basic equation (without damping) we want to solve as a system