Question

A pendulum of length 𝑙 at an angle 2𝛼. Find the time period T of the pendulum. Also let 𝛼 → 0 and obtain the well-known formula 𝑇 = 2𝜋√ 𝑙 /𝑔 .

Answer 1

$F = ma \\ m{a_x} = - T\sin \theta \\ \text{If } \sin \theta \sim \theta \\ m{a_x} = - T\theta \\ \text{Taking } \theta = \frac{x}{l} \\ m{a_x} = - T\frac{x}{l} \qquad (i) \\ m{a_y} = T\cos \theta - mg \\ \text{If } {a_y} = 0, T = mg \\ \begin{array}{l} {a_x} = \frac{{ - gx}}{l} = - {\omega ^2}x\\ \Rightarrow \omega = \sqrt {\frac{g}{l}} \\ T = \frac{{2\pi }}{\omega } = 2\pi \sqrt {\frac{l}{g}} \end{array} \\ \text{If } \theta = 2\alpha ,T = 2\pi \sqrt {2\alpha }$