Bucketta

Bucketta, a bucket-spinning water-witch of the Paradox Parlour, sets the water turning, and its surface climbs the walls into a curve.

The Physics

In the Scholium of the Principia (1687), Isaac Newton described a bucket of water hung on a twisted rope. When the bucket starts spinning, the water at first stays flat. As the water picks up the rotation, its surface becomes concave, and it stays curved even after the bucket is stopped, while the water keeps turning. Newton argued that the curve shows rotation relative to absolute space, not relative to the bucket.

z = ω²r²/(2g)

• z: height of the surface above its lowest point, on the axis (m)
• ω: angular speed of the water (rad/s)
• r: distance from the axis (m)
• g: gravitational acceleration (m/s²)

For water turning rigidly, the surface is a paraboloid: in the rotating frame, gravity and the centrifugal term together make a surface of constant pressure. Spinning liquid-mercury mirrors in telescopes use the same shape.

Honest caveat: The physics is settled; the meaning is not. Ernst Mach argued that the rotation might be relative to the distant mass of the Universe rather than to absolute space. General relativity includes some Mach-like effects, such as frame dragging, but does not fully realise Mach's principle, and how to interpret rotation remains a question in the philosophy of physics. The formula needs rigid rotation; in a still-accelerating bucket the shape is not yet a paraboloid.

On the sticker

Bucketta pulls the rope while a cyan whirl of spinning water glows in front of her. Printed text: BUCKETTA with the line z = ω²r²/(2g). Pixel-art kiss-cut vinyl sticker, white, in 2″, 3″, 4″ and 6″ sizes.