I Time Period of Vertical Circular motion

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1. Mar 20, 2016

Abhijeet Verma

Since the latex is not appearing, I request you to please click https://brilliant.org/discussions/thread/time-period-of-vertical-circular-motion/ , to view with complete formatting.
Thanks.
P.S-This is not schoolwork or homework.

Note: $$x$$ has been used for the angle with the vertical, measured in anticlockwise direction.
As shown in the figure, the tangential acceleration $${ a }_{ t }$$ is $$gsinx$$ .
Thus, the angular acceleration will be $$\frac { { a }_{ t } }{ R }$$ , where $$R$$ is the radius of the circle.
Writing
$\frac { \omega d\omega }{ dx} =-\frac { gsinx }{ R } int$
$\int _{ \frac { v }{ R } }^{ \omega }{ \omega d\omega } =\frac { g\int _{ 0 }^{ x }{ sinxdx } }{ R } \\ { This\quad gives\\ \omega =\sqrt { \frac { 2g(1-cosx) }{ R } +{ \frac { v }{ R } }^{ 2 } } =\frac { dx }{ dt } }$
Now, I don't know how to integrate this expression between $$0\quad to\quad 2\pi$$, to calculate the time taken for complete oscillation.
So, plese help by proceeding from here or if there is any other method to calculate the time period, please mention.

Thanks.

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2. Mar 20, 2016