Simple pendulum

What is it about?

The restoring force $F_r = m g \sin\varphi$ pulls the bob back to equilibrium. For small amplitudes ($\sin\varphi \approx \varphi$) the motion is harmonic with

The period depends only on the length and on $g$ – not on the mass. Since $T^2 = \frac{4\pi^2}{g}\, l$, a graph of $T^2$ against $l$ is a straight line through the origin with slope $4\pi^2/g$. For large amplitudes the period gets longer: up to about $60^\circ$, $T \approx T_0 \,(1 + \varphi_0^2/16)$ with $\varphi_0$ in radians; the simulation itself solves the equation of motion numerically and computes the period exactly as $T = T_0 / \operatorname{AGM}(1, \cos\tfrac{\varphi_0}{2})$, more than $1\,\%$ above $22.8^\circ$ and $18\,\%$ at $90^\circ$. Above $90^\circ$ a string would go slack, so the simulation uses a rigid rod.

Try it

1. Drag the bob aside and let go. Watch the energy bars while the total stays the same.
2. Turn on the force arrows. Forces from 90°
3. Compare with the small-angle approximation. Graphs
4. Time 10 oscillations at several lengths and determine $g$. Measurements
5. Damping turns the phase-space path into a spiral. Damped

Tasks

Task 1: Seconds pendulum

How long must a pendulum be for $T = 2\,\text{s}$?

Task 2: On the Moon

Find $T_0$ of a $1\,\text{m}$ pendulum on the Moon ($g = 1.62\,\tfrac{\text{m}}{\text{s}^2}$).

Task 3: Speed and tension

$m = 0.5\,\text{kg}$, $l = 1\,\text{m}$, released from $20^\circ$: how fast is the bob at the bottom? What is the tension there when released from $90^\circ$?

Ideas for the classroom

• The measurement series (with random reaction time) prepares a real stopwatch experiment: why time many periods, why use a small amplitude?
• The exact equation $\ddot\varphi = -\tfrac{g}{l}\sin\varphi - 2\gamma\dot\varphi$ is solved numerically; the small-angle approximation runs alongside as a dashed pendulum.
• Common stumbling blocks: “A heavier pendulum swings faster.” – “The tension always equals the weight.” (At the bottom it is larger by $m v^2 / l$.)