Conical Pendulum

Conical Pendulum is a small body suspended from a rigid support with the help of a string and revolving in a horizontal circle.

Conical Pendulum GIF

Consider a small body of mass $m$ suspended from a rigid support with the help of a string of length $l$. Suppose the body is moving in a horizontal circle of radius $r$. Let the string subtends an angle $θ$ with the vertical.
​Then, \[\sin θ=\frac{r}{l}\] \[r=l\sin θ\]

Derivation of the Time Period of a Conical Pendulum

In position $A$, weight of the body acts vertically downwards and tension $T$ acts along $AB$. Tension $T$ can be resolved into two components; $T\cos θ$ and $T\sin θ$.

$T\cos θ$ balances the weight of the body.
\[T\cos θ=mg\]
And, $T\sin θ$ provides the necessary centripetal force that enables it to move in a horizontal circular path.
\[T\sin θ = \frac{mv^2}{r}\]
Dividing above equations,
\[\tan θ = \frac{v^2}{rg}\] \[v=\sqrt{rg\tan θ}\] \[ωr=\sqrt{rg\tan θ}\]

If $T$ is the time period of the conical pendulum, then,
\[ω=\frac{2π}{T}\] \[∴ \frac{2π}{T}\cdot r = \sqrt{rg\tan θ}\] \[T=\frac{2πr}{\sqrt{rg\tan θ}}\] \[T=2π \sqrt{\frac{r}{g\tan θ}}\] Here, \[r=l\sin θ\] \[∴T=2π \sqrt{\frac{l\sin θ}{g \frac{\sin θ}{\cos θ}}}\] \[T=2π \sqrt{\frac{l\cos θ}{g}}\]
​This equation gives the time period of the conical pendulum.


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