Orbit Gate

The tether pulls straight at the hub, so it exerts no torque about the hub and angular momentum L = m v r is conserved. Reeling in shortens r, so v = L/(m r) rises — the craft genuinely speeds up as you pull.

The tension needed to hold the circle is F = m v²/r, and substituting v = L/(m r) gives F = L²/(m r³). Halving the radius multiplies the load by eight, which is why the last few metres cost so much. Setting F equal to the tether limit gives the snap radius drawn in red, the same relation as the safe-speed ceiling v_max = √(F_max r/m).

The dashed ghost line ahead of the craft runs the same update law forward — watch it dive and turn red while you hold, and relax outward when you release. Every frame the model advances the angle by ω Δt with ω = L/(m r²), which is exactly how a game engine integrates circular motion: keep the conserved quantity, derive the rest.