[NOTES/ME-06003]-Time Period of Oscillations--Method-1

1. Period of Oscillations-- Method-1

The total energy being a sum For a a particle having energy \(E\), and having bounded motion confined between two turning points, \(x_1\) and \(x_2\), the time period of oscillations will given by \begin{equation} T = 2\int_{x_1}^{x_2} \frac{dx}{\sqrt{\frac{2}{m}(E-V(x))}}. \end{equation} This is an exact result. If the integral can cannot be computed, it can be used to give approximate answer for the time period. For example, for a simple pendulum \begin{equation} V=mgL(1- \cos\theta). \end{equation} and the the time period for small amplitude \(\theta_0\) works out to be \begin{equation} T \approx 2\pi \sqrt{\frac{L}{g}}\Big(1+ \frac{\theta_0^2}{16}\Big). \end{equation}