Unit 05

Pendulum motion (a mass swinging in a gravitational field)is an obvious example of harmonic motion. The swing is periodic and the equation of motion can be calculated exactly, but the exact solution is very complicated. To avoid this complication, an approximation can be made which will allow you to write the pendulum equation of motion as Simple Harmonic Oscillator (SHO).

The full equation of motion is given as:

\[\frac{d^2\theta}{dt^2} = -\frac{g}{\ell} \sin(\theta)\]

Recall, the SHO equation is takes the following form \[\frac{d^2\theta}{dt^2} = -\frac{g}{\ell} \theta\]

The transformation between the two equations uses the approximation that \(\sin(\theta) \approx \theta\). Lets take a closer look at this approximation.

As you can see above, when the angle is small, the sine function (shown in red) looks almost exactly like a line (shown in blue)! As the angle gets larger, the approximation becomes less and less accurate as the angle increase. For simple experiments which examine pendulum motion in this course, the approximation breaks down around \(20^\circ\). Lets zoom in on this region and look only at the relative (percent) error for small angles

\[\text{Percent Error} = \left| \frac{\sin(\theta) - \theta}{\sin(\theta)} \right|\]

Please take a close look at the above figure. It shows the small angle approximation makes an error of approximately 2% at a release angle \(20^\circ\), and this error quickly increases to around 5% at \(30^\circ\). at \(40^\circ\), the error in the approximation is approaching 10%, and the SHO equation with simple solutions (\(\omega = \sqrt{g/\ell}\)) will not be true. In order to obtain accurate results at these larger angles, you are forced to use the more advanced mathematics which includes the full \(\sin(\theta)\) term.

What is the maximum allowable angle when using the small angle approximation? Unfortunately, I cannot give a definitive answer to this question. For simple pendulum measurements in this class, the effects of the approximation starts to become apparent around \(20^\circ\), but a more precise experiment would be able to measure the effect at even smaller angles. In the end, it is left up to the experimenter to determine what angle is acceptable for the given experiment.