Alright Guv'nor!

The other night, we watched the movie Mortal Engines, about a post-apocalyptic society that lives in steampunk-style mobile cities:

YouTube

All the retro-futuristic equipment got me thinking about one of the more popular devices to show, centrifugal governors.

Wikipedia

Some of the steam driving an engine is diverted into this device to make it rotate. The faster it goes, the more the arms swing outward. Depending on the angle, more or less steam is fed into the engine. I was curious to look at some of the physics behind these tools.

We can start by looking at just one of the arms:

The rotational inertia for this design is

which gives the rotational kinetic energy

Meanwhile, the height of the weight is

which gives the gravitational potential energy

We can combine those equations to relate ω and θ, and then use it in the torque equation:

In the case of a steam engine, we want to release pressure if the engine is going too fast, so we could imagine making the torque proportional to the height of the weights by connecting a valve to the arms. Then we get a differential equation for the rotation speed:

where A is the constant of proportionality. If A is negative, then high speeds get slower. As is, this equation would drive toward zero, but it could be engineered to have some minimum speed. I recognize it can get a bit silly at times, with useless gears and cumbersome designs, but I still have a soft spot for steampunk aesthetics, so I enjoyed the film.