# Time Period of Vertical Circular motion

Note: $$x$$ has been used for the angle with the vertical, measured in anti-clockwise direction.

As shown in the figure, the tangential acceleration $${ a }_{ t }$$ is $$g\sin x$$ . Thus, the angular acceleration will be $$\frac { { a }_{ t } }{ R }$$ , where $$R$$ is the radius of the circle. Writing $\frac { \omega d\omega }{ dx} =-\dfrac { g\sin x }{ R } int$ $\int _{ v/R }^{ \omega }{ \omega \, d\omega } =\dfrac { g\int _{ 0 }^{ x }{ \, sinx\, dx } }{ R } \\ { This\quad gives\\ \omega =\sqrt { \dfrac { 2g(1-\cos x) }{ R } +{ \dfrac { v }{ R } }^{ 2 } } =\dfrac { dx }{ dt } }$

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.

Thanks.

Note by Abhijeet Verma
2 years, 7 months ago

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

• bulleted
• list

1. numbered
2. list

1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

> This is a quote
This is a quote
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in $$...$$ or $...$ to ensure proper formatting.
2 \times 3 $$2 \times 3$$
2^{34} $$2^{34}$$
a_{i-1} $$a_{i-1}$$
\frac{2}{3} $$\frac{2}{3}$$
\sqrt{2} $$\sqrt{2}$$
\sum_{i=1}^3 $$\sum_{i=1}^3$$
\sin \theta $$\sin \theta$$
\boxed{123} $$\boxed{123}$$

Sort by:

There is no simple method to evaluate that integral.
Your equation is simply $$\dfrac{d^2 \theta}{d t^2} + \dfrac{g \theta}{R} = 0$$
The solution of this requires an elliptic integral to be solved.

- 2 years, 6 months ago