The Simple Pendulum

Truth is ever to be found in simplicity, and not in the multiplicity and confusion of things.Isaac Newton

A common example of a mechanical system that exhibits harmonic motion is the simple pendulum. A simple pendulum is an idealized system consisting of a bob of mass $m$ attached to a piece of string of length $L$. The equation of motion of this system is

$\displaystyle{ \dot{\theta}^2 =\frac{2 g}{L} \cos \theta + C \text{,} }$

where $C$ is a constant that is determined by the initial conditions of the pendulum. Two things to note about the differential equation above: 1) It is independent of mass $m$, and 2) it contains a useful parameter $g/L$ that the program will make use of as g_over_L. It is perhaps not surprising that the equation of motion is independent of mass since a simple laboratory experiment will reveal that altering the mass of the bob will have no effect on the period $T$ of the pendulum (more on period in the Notes). Moreover, we can encode this fact into the algorithm by using Newton's second law in "reduced units" with a(i) = f_osc. This also has the advantage of reducing the number of operations the simulation needs to perform.

The default parameters of this program generate a stable numerical solution when simulating large amplitude oscillations. Note that energy is given in reduced units of s^-2.

Reference

Boas, Mary. Mathematical Methods in the Physical Sciences. Wiley: John Wiley & Sons, 1983. 478.

Oscillations of a Pendulum

This program solves the equation for the total energy of a simple pendulum to illustrate conservation of mechanical energy for large oscillations.

$\displaystyle{ \frac{1}{2} m L^2 \left( \frac{\mathrm{d} \theta}{\mathrm{d} t} \right)^2 = m g L (\cos \theta - \cos \theta_0) }$

Initial angle $\theta_0$ of the mass: degrees

Length $L$ of the pendulum: m

Initial angular velocity $\omega_0$ of the mass: deg/sec

Time-step $\mathrm{d} t$ of the system: sec

Period $T$ of the pendulum: sec

Notes

gfortran -g -std=f2018 -Wall -Wextra -O2 -o amp amp.f08 • amp in Scratch

As this program illustrates, the period $T$ of the pendulum tends to increase slightly for a larger initial angle $\theta_0$. The prediction of this period, given by $T \approx 2 \pi \sqrt{L/g}$, is valid only for small angles ($\theta_0 \ll$ 57°), and so breaks down for $\theta_0 \simeq$ 23°. Of course, as this program simulates neither a damped nor driven harmonic system, the pendulum oscillates between $\theta_\text{max} = \pm \theta_0$.

Perhaps not surprisingly, the period of a simple pendulum can be generalized for a modular angle $\theta_0 \le$ 90° through a complete elliptic integral of the first kind $K(k)$:

$\displaystyle{ \begin{align*} T & = 4 \sqrt{\frac{L}{g}} K \left( \sin \frac{\theta_0}{2} \right) \\ & = 2 \pi \sqrt{\frac{L}{g}} \left[ \sum _{n=0}^{\infty} \left( \frac {(2n)!}{2^{2n} (n!)^2} \right)^2 \sin^{2n} \frac{\theta_0}{2} \right] \text{,} \end{align*} }$

where the elliptic modulus in this case is $k \equiv \sin \frac{\theta_0}{2}$.