• Celtrix Rattleback pixel-art physics T-shirt, white tee, front view

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Celtrix

The Rattleback Physics

\[\frac{d\mathbf{L}}{dt} = \mathbf{r}_c \times \mathbf{F}_c\]
$29.99 USD
$29.99 USD
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Size guide S–5XL · inches

Spin Celtrix the wrong way and she wobbles, rattles, and spins back the other way.

The Physics

A rattleback is a boat-shaped top whose mass axes are slightly twisted relative to its curved base (δ ≠ 0). Friction at the contact point couples wobbling to spinning, so in the unfavoured direction spin turns into rocking and then back into spin the other way. Angular momentum is not violated: the friction force at the contact exerts the torque.

dL/dt = r_c × F_c

  • L: angular momentum (kg·m²/s)
  • t: time (s)
  • r_c: position of the contact point from the centre of mass (m)
  • F_c: contact force from the table (N)
  • δ: misalignment angle (rad)

G. T. Walker analysed it in 1896.

Honest caveat: Some celts reverse more than once; not every rattleback reverses in only one direction.

On the shirt

Front: Celtrix, the pixel-art character, with the title. Back: the torque equation, the misalignment angle, the friction explanation and Walker's 1896 paper. Printed on a unisex tee.

Shipping calculated at checkout by destinationMade to order, ships in 2–5 business days

Details

Size & fitS–5XL

Unisex heavy cotton (Gildan 5000), classic fit.

SizeWidthLengthSleeve
S182815.1
M202916.5
L223018
XL243119.5
2XL263221
3XL283322.4
4XL303423.7
5XL323525

Zum Größenvergleich: Größe M ist 20 in (50.80 cm) breit, etwa 3.143 × 1034 Planck-Längen (lP = 1.616255 × 10−35 m). Ein Maßband geht schneller.

Measurements in inches, ±1.5 in tolerance.

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Shipping calculated at checkout by destination. Printed to order for you.

Delivery times & where we ship

Returns & replacements21 days

Every item is printed to order, so we can’t accept returns or exchanges for change of mind, or if you ordered the wrong size or colour.

Print defect, misprint, damage or the wrong item? We’ll send a free replacement, or a refund if you prefer. Report it within 21 days of delivery with a photo; no need to send it back.

Full refund & replacement policy

The physics

Spin me one way and I turn smoothly. Spin me the other way and I start to rock, slow to a stop, and spin back. People call me a celt or a rattleback, and I look as if I am breaking the conservation of angular momentum.

I am not. My mass axes are slightly twisted relative to the curve of my base, by an angle δ that is not zero. The table touches me at one point, and the force there, the normal force plus friction, exerts a torque r_c × F_c about my centre of mass. That torque turns spin into rocking in one direction, and rocking back into spin the other way.

G. T. Walker worked out the mathematics in 1896. Angular momentum is conserved for me plus the table plus the Earth; my own angular momentum changes because the table pushes on me. Some rattlebacks reverse more than once, and not every one reverses in only one direction.

Equations

\[\frac{d\mathbf{L}}{dt} = \mathbf{r}_c \times \mathbf{F}_c\]
SymbolMeaningUnit
\(\mathbf{L}\) angular momentum about the centre of mass \(\mathrm{kg·m²/s}\)
\(t\) time \(\mathrm{s}\)
\(\mathbf{r}_c\) vector from the centre of mass to the contact point \(\mathrm{m}\)
\(\mathbf{F}_c\) contact force from the table (normal force plus friction) \(\mathrm{N}\)
\(\delta\) skew angle between the principal axes of inertia and the axes of curvature at the contact (δ ≠ 0) \(\mathrm{rad}\)

Sources

  1. Garcia & Hubbard (1988) Spin reversal of the rattleback: theory and experiment, Proc. R. Soc. Lond. A 418, 165
  2. Kondo & Nakanishi (2017) Rattleback: A model of how geometric singularity induces dynamic chirality, Phys. Rev. E 95, 062207

Field notes

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