Torsional Spring Calculator

Background.

A torsional spring is Hooke's law bent into a circle: twist it through an angle θ and it pushes back with a torque T = κθ, proportional to the twist. The constant κ — the torsional spring rate, in newton-metres per radian — characterises everything from a clothes-peg's grip to a garage door's counterbalance, a mousetrap's snap, and the hairspring pacing a mechanical watch.

This page solves the relation in any direction: enter two of torque, rate, and angle, choose which to solve for, and it returns the third. Rearranged, the same line answers the three questions that actually arise — what torque does a known spring give at a known twist (T = κθ), what rate must a spring have to deliver a target torque within an allowed twist (κ = T/θ), and how far will a load twist a given spring (θ = T/κ).

The unit trap sits in the angle. The linear law is honest only in radians — κ's natural units are N·m/rad — but catalogues and drawings habitually quote degrees, and some list rates per degree or even per full turn. This calculator takes degrees at the interface and converts by π/180 internally; a rate quoted per degree must be multiplied by 57.3 to reach N·m/rad before it is entered. Getting this factor wrong produces errors of 57×, which is why it is the first thing to check when a spring calculation disagrees with a bench measurement.

The model is the ideal linear spring about its working point — exactly what spring makers specify within the working range. Real springs add preload, rate tolerance (±10% is common), friction between coils and over arbors, and a maximum safe deflection; deriving κ itself from wire diameter and coil geometry is a separate design calculation, as the scope note beside the result records.

What is torsional spring calculator?

A torsional spring resists angular deflection with a restoring torque proportional to the twist angle: T = κθ, the rotational analogue of Hooke's F = kx. The torsional rate κ (also called torsional stiffness) has units of torque per angle — properly N·m/rad — and doubles as the energy bookkeeper: a spring twisted by θ stores U = ½κθ². The linear law describes helical torsion springs, spiral hairsprings, and torsion bars alike within their working range, with the twist measured from the spring's free position.

How to use this calculator.

1. 01Pick the unknown with ‘Solve for’: torque delivered, rate required, or deflection produced — the two relevant knowns are read and the third input is ignored.
2. 02Enter angles in degrees as the field is labelled; the π/180 conversion to radians happens inside, where the physics needs it.
3. 03Check catalogue rate units before entering κ: N·m/rad goes straight in, N·m per degree must be multiplied by 57.296 first, and ‘per turn’ rates divided by 2π.
4. 04Measure deflection from the free (unloaded) position — many installed springs carry preload, so the working torque is κ times the total twist from free, not from the installed position.
5. 05Keep the answer inside the spring's rated deflection: the linear law says nothing about coil bind, yield, or fatigue, which set the real limits.

The formula.

T = κθ is the small-deflection limit of elasticity applied to twisting, the exact rotational mirror of F = kx: torque replaces force, angle replaces displacement, and the rate κ replaces k. Radians are forced by the physics — the radian is the dimensionless angle that makes arc length equal radius times angle, so only in radians is κ a clean torque-per-angle without a hidden conversion constant. The relation integrates to the stored energy U = ½κθ² (the area under the torque-angle line), which is what a mousetrap releases and a watch hairspring meters out; it also sets oscillation frequency, ω = √(κ/I) for inertia I, the balance-wheel equation of horology. For a solid round torsion bar, κ derives from geometry and material as GJ/L — shear modulus times polar second moment over length — which is how designers hit a target rate; helical torsion springs have their own standard design formulas. The engine applies the degree-to-radian factor and one multiplication or division in Decimal arithmetic, rounding once to twelve significant digits.

A worked example.

Frequently asked questions.

References& sources.

1. [1]OpenStax, Rice University. University Physics, 2016. Volume 1 mechanics and Volume 2 thermodynamics. Retrieved 2026-08-06. independence: primary; access: open.
2. [2]NIST. SI derived units current 2026. Torque and plane-angle units. Retrieved 2026-08-06. independence: secondary-check; access: open.
3. [3]OpenStax, Rice University. Chemistry 2e, 2019. Chapters 4, 9, and 12. Retrieved 2026-08-06. independence: primary; access: open.