Pendulum Physics

In this project, understanding the physics of a simple pendulum is essential for analyzing its motion and estimating both gravitational acceleration and the minimum frame rate required for accurate tracking. This section outlines the fundamental principles of pendulum physics as applied to our setup and provides the formulas used to derive gravity and frame rate requirements.

Simple Pendulum Model

To model the pendulum’s motion, we start with the basic force equation for a mass $m$ suspended from a string of length $L$:

$$\sum F = m \cdot \ddot{x}$$

$$-m \cdot g \cdot \sin(\theta) = m \cdot \ddot{x}$$ $$-\sin(\theta) \cdot g = \ddot{x}$$

For small angles, we can approximate $\sin(\theta) \approx x / L$, where $x$ is the horizontal displacement. This leads to:

$$-\frac{g \cdot x}{L} = \ddot{x}$$

Solving this differential equation yields:

$$x(t) = A \cdot \sin\left(\sqrt{g/L} \cdot t + \phi\right)$$

where:

• $A$ is the amplitude of the oscillation,
• $\phi$ is the phase constant, depending on initial conditions,
• $\frac{g}{L}$ represents the square of the angular frequency $\omega^2$.

Using the relationship $\omega = \frac{2\pi}{T}$, we find the period $T$ of the pendulum:

$$T = 2\pi\sqrt{L/g}$$

Estimating Gravity

From this, gravitational acceleration $g$ can be estimated with the average measured period $T_{avg}$:

$$g = 4\pi^2\frac{L}{T_{avg}^2}$$

where:

• $T_{avg}$ is the average measured period (in seconds),
• $L$ is the pendulum length (in meters).

This gives $g$ in $\text{m/s}^2$ when all units are in the International System.

Determining the Required Frame Rate

According to the sampling theorem:

A signal can be accurately reproduced if it is sampled at a frequency $F$ that is greater than twice the maximum frequency of the signal.

The maximum frequency of the pendulum’s motion corresponds to the system's natural frequency:

$$f_{max} = f = \frac{1}{T} = \frac{1}{2\pi}\sqrt{g/L}$$

Thus, our sampling frequency $F_{sample}$ should be at least: $$F_{sample} > \frac{1}{\pi}\sqrt{g/L}$$

To capture the motion more reliably, a common rule of thumb suggests sampling at 10 times the signal’s maximum frequency. Therefore, we should aim for:

$$F_{sample} \approx \frac{5}{\pi}\sqrt{g/L}$$

For $L = 0.7 \text{m}$ and $g \approx 10 \text{m/s}^2$, the required frame rate is approximately $6 \text{Hz}$ or 6 fps, which is achievable even with relatively low-optimized embedded software.

References

• The Physics Hypertextbook - Pendulum
• Wikipedia - Pendulum
• Wikipedia - Sampling Theorem