Revisitando o Experimento de Erat\'ostenes: medida do raio de Terra

TL;DR

This paper replicates Eratosthenes' ancient experiment using simple tools in Brazilian cities to measure Earth's radius and angular velocity, achieving high accuracy and demonstrating educational methods for planetary measurement.

Contribution

It presents a practical, low-cost method to estimate Earth's radius and angular velocity using basic instruments and modern satellite data, revisiting a historical experiment with contemporary validation.

Findings

Earth's radius estimated with 0.5% error

Method demonstrates educational value and simplicity

Angular velocity of Earth also estimated

Abstract

In this work, we measure the volumetric mean radius of the Earth, reproducing the historical experiment of Eratosthenes, carried out around 240 BC, in the ancient cities of Siena and Alexandria. Here, we perform measurements in the cities of Rio de Janeiro-RJ-Brazil and Teresina-PI-Brazil, whose longitude coordinates are close. Using simple equipment, such as plumb lines and rulers, we simultaneously measure the height of an object and its shadow's length at midday, when the Sun is at its highest elevation, in order to obtain the shadow angle. After determining the distance (latitude) between cities from satellite data, we use the measured shadow angles to estimate the volume mean radius of the Earth, from basic trigonometric arguments, finding a result whose error is $0.5 \% $ with respect to the literature. Furthermore, from the difference of time in which the Sun is at the highest point in the sky in each location, we also estimate the Earth's angular velocity. In summary, we pedagogically present how to obtain the mean curvature of the planet, assuming an approximately spherical surface, in addition to its angular velocity.