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13.4: The Motion of a Pendulum
https://phys.libretexts.org/Courses/Berea_College/Introductory_Physics%3A_Berea_College/13%3A_Simple_Harmonic_Motion/13.04%3A_The_Motion_of_a_Pendulum
However, consider the “the small angle approximation” 1 for the sine function: $\begin{aligned} \sin\theta \approx \theta \end{aligned}$ If the oscillations of the pendulum are “small”, such that th...However, consider the “the small angle approximation” 1 for the sine function: $\begin{aligned} \sin\theta \approx \theta \end{aligned}$ If the oscillations of the pendulum are “small”, such that the small angle approximation is valid, then the equation of motion for the pendulum is: $\begin{aligned} \frac{d^2\theta}{dt^2} &= -\frac{g}{L}\sin\theta \approx -\frac{g}{L}\theta \\ \therefore \frac{d^2\theta}{dt^2} &=-\frac{g}{L}\theta \quad (\text{for small }\theta)\end{aligned}$ and the angle…

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