Estimation of $π$ via experiment

TL;DR

This work experimentally demonstrates a collision-counting method to estimate $π$ by utilizing two bodies and a wall with a mass ratio of $1:100$, employing a suspended apparatus to minimize energy losses and achieve 31 collisions, which corresponds to $π$ to the tenths place ($3.1$). The authors develop a velocity-space framework where conservation of momentum and energy constrains the state to intersections of a line and a circle in the rescaled plane, with collisions advancing the state by a fixed angular step around the circle. They validate the theory by achieving the theoretical collision count under realistic conditions and discuss the geometric versus dissipative origins of finite collisions, outlining a path toward higher precision (e.g., $1:10000$) that faces substantial engineering challenges. The study provides a tangible, teachable realization of a classic result and highlights the practical trade-offs between idealized geometry and real-world energy losses.

Abstract

In this study, we conducted an experiment to estimate $π$ using body-to-body and body-to-wall collisions. By geometrically analyzing the system's motion, we first review how the collision count corresponds to the digits of $π$. This method utilizes the property that the number of collisions corresponds to $π$ to the $n$-th decimal place by setting the mass ratio of bodies to $1:100^n$ under ideal conditions. In particular, when the mass ratio is $1:100$ -- which is the case we tested experimentally -- the number of collisions is 31, and $π$ to the tenths decimal place (3.1) can be derived. In the experiments, a suspended apparatus was developed to minimize energy losses such as friction and air resistance. We also devised the shape and material of the colliding bodies' surface and the characteristics of the suspension string, aiming for measurements under stable conditions. Based on the experimental results, we reproduced the number of collisions consistent with the theoretical values and confirmed that estimating $π$ to the tenths decimal place is possible under realistic conditions.

Figures (7)

• Figure 1: Schematic sequence of collisions between two bodies and a wall.
• Figure 2: Case 1 ($m\!:\!M=1\!:\!1$): Motion trajectory in the $V_{1}$–$V_{2}$ plane.
• Figure 3: Same as Fig. \ref{['fig:phasecircle1']}, but for Case 2 ($m\!:\!M=1\!:\!100$).
• Figure 4: Overview diagram of experimental apparatus.
• Figure 5: Photographs of the whole experimental apparatus. In the left figure, the wall has been removed so that the colliding bodies can be photographed.