What is a Conical Pendulum?
A conical pendulum consists of a bob of mass $m$ attached to a string of length $l$ which is fixed at one end. The bob moves in a horizontal circle such that the string traces the surface of a cone. Unlike a simple pendulum which swings back and forth in a plane, the conical pendulum maintains a constant vertical distance and traces a circular path.
Derivation of the Time Period
To find the time period $T$, we consider the forces acting on the bob:
- Tension ($T_s$) in the string.
- Weight ($mg$) acting downwards.
Let $\theta$ be the semi-vertical angle the string makes with the vertical. We resolve the tension $T_s$ into two components:
- Vertical component: $T_s \cos \theta = mg$ (to balance gravity)
- Horizontal component: $T_s \sin \theta = \frac{mv^2}{r}$ (providing centripetal force)
From these, we divide the two equations: $\tan \theta = \frac{v^2}{rg}$
Given the radius of the circular path $r = l \sin \theta$, we substitute this into the expression for $v$: $v = \sqrt{rg \tan \theta} = \sqrt{gl \sin \theta \tan \theta}$
Since $v = \frac{2\pi r}{T}$, we have $T = \frac{2\pi r}{v}$. Substituting $r$ and $v$: $T = 2\pi \sqrt{\frac{l \cos \theta}{g}}$
Intuition
Think of the conical pendulum as a system where horizontal motion is decoupled from vertical motion. The vertical height of the pendulum $h = l \cos \theta$ determines the time period, which is why the formula looks similar to a simple pendulum $T = 2\pi \sqrt{\frac{h}{g}}$.