Yo-Yo Physics 2D: String-Unwinding Rotational Dynamics

Yo-Yo Controls

Axle radius 3.0 mm

Moment of inertia 1.00×

String length 1.0×

Ready — press Drop to release the yo-yo.

Live Readout

Fall speed

0.00 m/s

Spin rate

0 rpm

String tension

0.00 N

Unwound

How it works

A yo-yo is a disc of outer radius R whose string is wound not around the rim but around a much thinner axle of radius r through its centre. Gravity pulls the whole yo-yo down with weight mg, but the string, anchored at the top, can only pay out at the axle — so every metre the yo-yo falls, the axle must also spin by that arc length, coupling the linear fall to rotation.

Newton's second law for the fall (mg − T = ma) and for the spin about the axle (T·r = I·α, with a = α·r) combine into a clean closed form:

a = g / (1 + I / (m·r²)) — the linear acceleration while unwinding. Because r is small, I/(m·r²) is large, so a is much smaller than g: a yo-yo falls slowly compared to a dropped ball, even though gravity is doing all the work.
String tension T = m(g − a) — the string carries only part of the yo-yo's weight; the rest goes into building up rotational kinetic energy.
A thinner axle (smaller r) or a larger moment of inertia I (mass pushed toward the rim) both slow the fall further and store more spin energy per metre dropped.

When the string fully unwinds, an ideal string could not keep applying torque — real yo-yos exploit a slack loop and friction at the axle (a crude centrifugal clutch): below a spin-speed threshold the loop tightens and grips, but above it the yo-yo "sleeps", spinning nearly freely at the bottom of the string while friction slowly bleeds off its rotational energy. A sharp upward tug on the string momentarily increases tension past the clutch's grip threshold, the loop catches the axle again, and the disc reverses to climb — mirroring the string-friction-clutch behaviour that makes "sleeping" and "the sleeper trick" possible on a real yo-yo.

Disc / axle String