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#pendulum

9 posts7 participants0 posts today
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Nolan Moore

Finally got #Pendulum to the table 382 days after purchase! Took a moment to get used to the unique gameplay, but it ended up being easy to understand and was a very enjoyable. Played a couple solo games and am really looking forward to introducing my group to its real-time* worker placement soon.

I also participated with Jamey Stegmaier’s live-streamed #RollingRealms game. Haven’t received my Tokaido realm promo yet, but I made do!

The Pendulum board game is setup on a table with a floral pattern tablecloth. The player’s board is closest to the photographer with the Automa board next to it, both surrounded by player tokens. The main board is further back and has an array of worker meeples on action spaces. Cards are missing as the game is being cleaned up.
A single player setup of Rolling Realms is set up on a floral table cloth. The player’s score card shows the dice rolls of all 9 rounds as well as scores for the 3 games (11.5, 13.7, 12.3, and a total of 37.5). The realms for Stamp Swap, Finspan, and Tokaido are in the foreground. Tokaido’s realm is drawn on a notepad in place of an official card.
#WhatDidYouPlayMondays#BoardGames
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Pustam | पुस्तम | পুস্তম🇳🇵

A cycloidal pendulum - one suspended from the cusp of an inverted cycloid - is isochronous, meaning its period is constant regardless of the amplitude of the swing. Please find the proof using energy methods: Lagrange's equations (in the images attached to the reply).

Background:
The standard pendulum period of \(2\pi\sqrt{L/g}\) or frequency \(\sqrt{g/L}\) holds only for small oscillations. The frequency becomes smaller as the amplitude grows. If you want to build a pendulum whose frequency is independent of the amplitude, you should hang it from the cusp of a cycloid of a certain size, as shown in the gif. As the string wraps partially around the cycloid, the effect decreases the length of the string in the air, increasing the frequency back up to a constant value.

In more detail:
A cycloid is the path taken by a point on the rim of a rolling wheel. The upside-down cycloid in the gif can be parameterized by \((x, y)=R(\theta-\sin\theta, -1+\cos\theta)\), where \(\theta=0\) corresponds to the cusp. Consider a pendulum of length \(L=4R\) hanging from the cusp, and let \(\alpha\) be the angle the string makes with the vertical, as shown (in the proof).

GIF
#Pendulum#Cycloid#Period
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