🌀 Torsion Pendulum — Angular Oscillator

Mass M and radius R set the disk's moment of inertia through I = ½MR². The torsion constant κ slider sets how strongly the wire resists twisting. The damping slider sets the angular damping coefficient c, which controls how quickly oscillations die out. The initial twist slider sets θ0, the angle the disk is released from at t = 0. The Release/Restart button resets the clock to zero with the disk held at θ0 and zero angular velocity.

Moment of inertia measures how hard it is to change a body's rotation rate, just as mass measures how hard it is to change its straight-line velocity. For a solid disk, I = ½MR², so doubling the mass only doubles I, but doubling the radius quadruples it, because mass farther from the spin axis contributes far more resistance to angular acceleration than mass close to it. That is why a wide, thin disk behaves very differently from a small, dense one of the same mass.

The damping ratio ζ = c / (2√(Iκ)) compares the actual damping to the specific amount that would make the system just barely fail to oscillate. If ζ is less than 1 there is not enough damping to prevent overshoot, so the disk oscillates with a slowly shrinking amplitude — underdamped. If ζ equals 1 the damping is exactly enough to return the disk to rest as fast as possible without any overshoot — critically damped. If ζ is greater than 1 there is more damping than necessary, and the disk crawls back to equilibrium even more slowly, still without oscillating — overdamped.

The natural period T0 = 2π√(I/κ) is how long one oscillation would take if there were no damping at all. The damped period Td = 2π/ωd, where ωd = ω0√(1−ζ²), is the actual period observed once damping is present, and it is always slightly longer than T0 because the restoring torque has to work against friction as well as inertia. As damping increases toward critical, Td stretches toward infinity and then the concept of a period stops applying altogether, which is why the telemetry panel shows "N/A" once oscillation ceases.

Underdamped systems return quickly but overshoot and oscillate, losing time to each swing past zero. Overdamped systems never overshoot but move sluggishly because the large damping force also opposes the useful part of the motion. Critical damping sits exactly at the boundary: it supplies just enough resistance to prevent any overshoot while still allowing the fastest possible approach to equilibrium, which is why engineers deliberately design dampers — in door closers, suspension systems and instrument needles — to sit as close to critical as possible.

Henry Cavendish used a torsion balance to measure the gravitational constant and effectively "weigh the Earth" in 1798, and the same principle underlies modern torsion balances used in precision gravity experiments. Mirror galvanometers used torsion fibres to detect tiny electrical currents by rotation of a mirror, mechanical wristwatches and clocks have used torsion-based balance wheels and pendulums, and some modern MEMS gyroscopes and resonant sensors rely on the same underlying rotational spring-mass-damper physics modelled here.

Changing the mass, radius, torsion constant, damping or initial angle changes the underlying differential equation itself — a new moment of inertia, a new natural frequency, or a new damping ratio. Rather than splice a changed equation into a mid-flight motion, the simulator restarts cleanly from θ0 with zero angular velocity so the trajectory you see always corresponds exactly to the parameters currently shown in the telemetry panel.