Three pendulums with strings of the same length and bobs of the same mass are pulled are pulled out to angles $\theta_1, \theta_2 \sf\ and\ \theta_3$ respectively and released. The approximation $ sin\ \theta = \theta$ holds for all three angles, with $\theta_3 \theta_2 \theta_1$. How do the angular frequencies of the three pendulums compare?

why would they be equal? ? just need an explanation.

Question

Three pendulums with strings of the same length and bobs of the same mass are pulled are pulled out to angles $\theta_1, \theta_2 \sf\ and\ \theta_3$ respectively and released. The approximation $ sin\ \theta = \theta$ holds for all three angles, with $\theta_3 > \theta_2 > \theta_1$. How do the angular frequencies of the three pendulums compare?

why would they be equal? ? just need an explanation.

anonymous · Answer

@texaschic101 @IrishBoy123 @KendrickLamar2014

irishboy123 · Answer

good question normally, we'd use $\Large T = 2\pi \sqrt{L\over g} $ for the period from the small angle approximation, and so it's all about L and g. but then I found this on wiki: $\Large T = 2\pi \sqrt{L\over g} \left( 1+ \frac{1}{16}\theta_0^2 + \frac{11}{3072}\theta_0^4 + \cdots \right)$ so for **all** angles the period is that wee bit bigger for a bigger initial displacement, which makes sense as the distance travelled is clearly greater. but for small swings it's hard to tell the difference. more detail here plus some nice animations too https://en.wikipedia.org/wiki/Pendulum the snazzy maths for that longer equation is here, but i think you are more interested in the why https://en.wikipedia.org/wiki/Pendulum_(mathematics)#Arbitrary-amplitude_period