Aristotle's wheel paradox: a small inner circle seems to roll farther than its own circumference

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Aristotle's wheel paradox: a small inner circle seems to roll farther than its own circumference

the gist

Take a wheel made of two concentric circles fixed together, with the large outer circle rolling along a flat surface. When the wheel turns exactly once, the outer circle covers a distance equal to its circumference. The inner circle, which is smaller, seems to cover that same distance, even though its circumference is shorter. The puzzle appears in the pseudo-Aristotelian work Mechanica. The paradox is resolved by seeing that the inner circle does not simply roll: because it is fixed to the outer one and moves with its centre, its contact points slide forward along the ground. The same effect appears with rolled tape, bottles and jars.

how it works

A rigid pair of circles shares one centre, so both move forward together by the same amount. The outer circle touches the ground without slipping, so its circumference matches the distance. The inner circle is carried along while its contact point is forced to slide along the surface, adding extra ground distance. Each point of the inner circle traces a looping path, which makes the total path longer than its circumference.

so what

The puzzle shows that rolling motion is a combination of translation and rotation, and that a rotation count does not directly tell you distance travelled for a rigid body that is not in pure contact. It has been a classic lesson in mechanics for centuries.

the bigger pattern

The same paradox returns in train wheels with flanges, where the smaller circle touches the rail. The Israeli educator and philosopher Israel Drabkin named these as Case II versions. Similar puzzles appear in the history of mechanics from Galileo onward.

fun fact

The Mechanica is attributed to Aristotle but was written by someone else.

Hampi was once one of the world's largest cities, until its Vijayanagara capital was destroyed in 1565

the gist

Hampi is a UNESCO World Heritage Site in Karnataka, India, with more than 1,600 surviving remains spread over 4,100 hectares. It was the capital of the Vijayanagara Empire from 1336 to 1565. Travellers from Persia and Portugal described it as a prosperous, wealthy and grand city with temples, markets and farms along the Tungabhadra River. By 1500 it was estimated to be the world's second-largest city after Beijing, and probably India's richest at the time. In 1565 a coalition of Muslim sultanates defeated the Vijayanagara Empire, its capital was conquered and destroyed, and the city was left in ruins. Hampi remains a religious centre, with the active Virupaksha Temple at its heart.

how it works

The city was built around a fortified core with temples, markets and water structures along the river, supplied by an extensive network of roads and waterworks. When the empire fell, the city was not rebuilt, so its stone monuments were left standing.

so what

Hampi shows how quickly a great city can vanish after a military defeat, and how a ruined capital can remain sacred. Its ruins give historians a view of medieval South Asia that few other sites can match.

fun fact

The name Hampi comes from Pampa, an old name for the Tungabhadra River.

From Wikipedia, the free encyclopedia · text CC BY-SA 4.0

Image: Wikimedia Commons

source: Wikipedia ↗

2 min

politicsshortCNA2:01

False allegations about Singapore’s hand in Indonesia’s markets and a made-up meeting

from CNA

CNA Verify has found close to 400 posts pushing falsehoods tying Singapore to Indonesia’s stock market rout and to a secret meeting that never took place. Together, they drew nearly 12 million views, and some were circulated by accounts that appeared to be working in concert, across multiple social media platforms.

Here’s how the disinformation campaign targeting Singapore was uncovered.

on YouTube: CNA ↗