Conical Pendulum Period — Does It Ever Match a Simple Pendulum of the Same Length?

Conical Pendulum: How the Cone Angle Sets Period, Tension and Speed

Conical Pendulum: How the Cone Angle Sets Period, Tension and Speed

Seen from above at an angle, so an arrow pointing toward or away from you looks shorter. COMPONENTS adds the two parts of FT, dashed: FT cos β straight up and FT sin β toward the centre.
Plot
Cone angle β
String length ℓ
Mass m
PHYSICS INSIGHTS

The string’s pull does two jobs. Its vertical part holds the bob up: FT cos β = mg. Its horizontal part is the whole net force, pointing at the centre of the circle: FT sin β = mv2/r. Divide one by the other and the mass cancels: tan β = v2/(gr), so ac = g tan β. Mass never appears in β, T, v, ac or r; it only scales the tension.

Wider cone, faster orbit, shorter period. With r = ℓ sin β and T = 2π√(ℓ cos β ÷ g), opening the cone widens the circle yet shortens T: v climbs without limit while T falls toward zero. A heavy bob and a light one on the same string at the same angle take exactly the same time per revolution.

Why 90° is impossible. As β nears 90°, cos β nears 0, so FT = mg ÷ cos β grows without limit: a level string has no vertical part left to hold up the weight. Watch FT ÷ mg in the readout: it reaches 2 at 60°, and at 85°, where this sim stops, the tension is about 11.5 times the weight.

Simulation by The Science Cube — https://www.thesciencecube.com/
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