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

Teachers: assign this simulation to your class in one click. Free for classroom use. Share to Google Classroom
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 center.
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 center 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/

Does a conical pendulum ever have the same period as a simple pendulum of the same length?

In short: Only in the limit of a vanishing cone angle. A conical pendulum has period T = 2π√(ℓ cos β ÷ g). Because cos β is less than 1 for any open cone, T is always a little shorter than the small-swing simple pendulum value 2π√(ℓ/g), and widening the cone shortens it further.

Common doubts

  • “Does a heavier bob change the period?” No. Mass cancels out of tan β = v²/(gr). A heavy and a light bob on the same string at the same angle take exactly the same time per revolution. Mass only scales the tension.
  • “Why does a wider cone give a shorter period?” The circle gets wider, r = ℓ sin β, but the bob moves faster by more than enough to make up for it, so T falls while v climbs.
  • “Why can the string never be horizontal?” The vertical part of the tension must hold up the weight: FT cos β = mg. As β nears 90°, cos β nears 0, so FT = mg ÷ cos β grows without limit.

Change the cone angle, string length and mass in the simulation above and watch T, v and the tension respond.

Want more AP Physics 1 help like this?


Equations & Traps: the AP Physics 1 equation deck. See Equations & Traps

AP Physics 1 Mind Maps: every topic on one page. See the Mind Maps

Complete and Continue