Conical Pendulum — Theory, Formula and Derivation

Understand the conical pendulum theory formula and derivation with clear steps. A conical pendulum is a mass suspended from a fixed point that moves in a horizontal circular path, making a constant angle with the vertical.

What Is a Conical Pendulum?

A **conical pendulum** consists of a bob of mass m suspended by a string of length L that describes a **horizontal circle** with constant speed such that the string traces out a **cone**. The bob moves in uniform circular motion while the string makes a constant angle θ with the vertical.

Forces Acting on the Bob

There are two forces on the bob:

• Tension T in the string
• Weight mg (acting vertically downward)

The tension can be resolved into vertical and horizontal components.

Derivation (Step‑by‑Step)

Let the bob describe a horizontal circle of radius r = L sin θ and make an angle θ with the vertical.

The vertical component of tension balances weight:

T cos θ = mg

The horizontal component provides centripetal force:

T sin θ = m v² / r

Substitute T from the first equation:

(mg / cos θ) sin θ = m v² / (L sin θ)

g tan θ = v² / (L sin θ)

Final Formula

The speed of the bob:

v = √(g L sin θ tan θ)

The time period (Tₚ) of one revolution:

Tₚ = 2π √(L cos θ / g)

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Conical Pendulum — Theory, Formula and Derivation

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Conical Pendulum — Theory, Formula and Derivation