Conical Pendulum Calculator
Calculate the period, angular velocity, speed, string tension and orbit radius of a conical pendulum from the string length, cone angle and bob mass.
How it works
T = 2π·√(L·cos θ / g) · ω = √(g / (L·cos θ)) · r = L·sin θ · v = ω·r · tension Fₜ = m·g / cos θ · centripetal force F꜀ = m·g·tan θ
In a conical pendulum the bob moves in a horizontal circle while the string sweeps out a cone. The vertical part of the tension balances the weight (Fₜ cos θ = mg) and the horizontal part supplies the centripetal force (Fₜ sin θ = mω²r). Dividing the two gives ω² = g/(L cos θ), so the period depends only on the height of the cone h = L cos θ, not on the mass. This is the standard uniform-circular-motion result in OpenStax University Physics and Serway & Jewett.
Worked example
A 0.5 kg bob on a 1 m string swings at 30° from the vertical (g = 9.81 m/s²). The cone height is h = 1 × cos 30° = 0.866 m, so T = 2π × √(0.866 / 9.81) = 1.867 s (32.14 rpm). The angular velocity is ω = 3.366 rad/s, the radius r = 1 × sin 30° = 0.500 m and the speed v = 3.366 × 0.5 = 1.683 m/s. The tension is 0.5 × 9.81 / cos 30° = 5.66 N and the centripetal force 0.5 × 9.81 × tan 30° = 2.83 N.
Sources: OpenStax University Physics Vol. 1 — Centripetal Force · HyperPhysics — Conical pendulum
Frequently asked questions
Does the mass of the bob change the period?
No. Mass cancels out of ω² = g/(L cos θ); it only scales the tension and the centripetal force.
How is a conical pendulum different from a simple pendulum?
A simple pendulum swings back and forth in a plane; a conical pendulum goes round in a horizontal circle at a constant angle. For small angles both have nearly the same period 2π√(L/g), but the conical period gets shorter as the angle grows.
What happens as the angle approaches 90°?
cos θ goes to zero, so the required speed and tension grow without limit — the string can never be exactly horizontal while supporting the weight.
Where is the conical pendulum used?
Watt’s flyball governor on steam engines, fairground swing rides, tetherball and lab demonstrations of centripetal force all behave like conical pendulums.
Related calculators
Results assume the idealised textbook model stated above; real systems have losses and tolerances, so verify safety-critical values with measurements or a qualified engineer.