What is a simple pendulum? Find an expression for the time period and frequency of a simple pendulum.

Explanation

The period of a conical pendulum is the time taken by its bob to complete one revolutio in a horizontal circle with constant speed.
Consider a conical pendulum of string length L with its bob of mass m performing UCM along a path of radius r.
The linear speed of the bob of a conical pedulum,
`v = sqrt(rg tan theta)`
If T is the periodic time of the UCM , the bob travels a distance of `2pir` in time T. So that,
`T = ("distance")/("speed") = (2pir)/(sqrt(rg tantheta))" "thereforeT = 2pisqrt((r)/(g tan theta))`
FROm the diagram, ` r = L sin theta` .
`therefore T = 2pisqrt((L sin theta)/(g tan theta))=2pisqrt((L costheta)/(g))`
This is the required expression. From the above expression, we can see that
(i) `T prop sqrt(L)`
(ii) `T prop sqrt(cos theta)` (if `theta` increases, `cos theta` and T decrease)
(iii) `T prop (1)/(sqrt(g))`
(iv) The period is idenpendent of the mass of the bob.
`[Note :L cos theta =h " "thereforeT = 2pisqrt((h)/(g))]`