• Kapitzar – Kapitza Pendulum physics T-shirt
  • Kapitzar – Kapitza Pendulum physics T-shirt
  • Kapitzar – Kapitza Pendulum physics T-shirt
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KapitzarKapitza Pendulum

Kapitzar – Kapitza Pendulum Physics T-Shirt

$29.99 USD
$29.99 USD
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White · Black
Size
S–5XL
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Can shaking make an upside-down pendulum stable? Yes, if the pivot vibrates up and down rapidly enough. Averaged over the fast shaking, the motion feels an extra effective force pushing it toward the upright position, so the inverted state becomes stable. Pyotr Kapitza analysed this effect. Kapitzar balances on that buzzing rod. Pixel-art unisex tee; equation, symbols and sources in the physics panel below.

Color
Size

Size guideS–5XL · inches

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

Measurements in inches, ±1.5 in tolerance.

ShippingFree US

Free US shipping · $10 flat shipping outside the US. 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

Balance a pendulum upside down and it falls. Vibrate its pivot up and down fast enough and it stands.

Averaged over the fast shaking, the pendulum feels an extra potential that pulls it towards the vertical. When the pivot's speed beats the square root of 2gl, the inverted position becomes a stable dip. Pyotr Kapitza explained it in 1951.

Kapitzar stands on a buzzing base. Buzz the base. I stand.

Equations

\[U_{\text{eff}}(\theta) = -m g l\cos\theta + \frac{m a^{2}\omega^{2}}{4}\sin^{2}\theta\]
\[a^{2}\omega^{2} > 2\,g\,l\;\;\Rightarrow\;\;\theta = \pi\ \text{(upside down) is stable}\]
SymbolMeaningUnit
\(U_{\text{eff}}\) time-averaged effective potential of the pendulum \(\mathrm{J}\)
\(\theta\) angle from the hanging-down position \(\mathrm{rad}\)
\(m\) mass of the bob \(\mathrm{kg}\)
\(g\) gravitational acceleration \(\mathrm{m/s^2}\)
\(l\) length of the pendulum \(\mathrm{m}\)
\(a\) amplitude of the pivot's vertical vibration \(\mathrm{m}\)
\(\omega\) angular frequency of the pivot vibration \(\mathrm{rad/s}\)

Sources

  1. Butikov (2001) On the dynamic stabilization of an inverted pendulum, Am. J. Phys. 69, 755
  2. Kapitza (1951) Dynamic stability of a pendulum when its point of suspension vibrates, Zh. Eksp. Teor. Fiz. 21, 588 (citation only)
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