Question

Three pendulums with strings of the same length and bobs of the same mass are pulled out to angles θ1, θ2, and θ3, respectively, and released. The approximation sin θ = θ holds for all three angles, with θ3 > θ2 > θ1. How do the angular frequencies of the three pendulums compare?

Answer 1

The angular frequencies of all the 3 pendulums shall be same.

Step-by-step explanation:

The time period of a simple pendulum with the approximation
`sin(\theta )\approx \theta`is given by:

`T=2\pi\sqrt{(l)/(g)}`

The angular frequency
`\omega` is given by

`\omega =(2\pi )/(T)\\\\\omega =\frac{2\pi}{2\pi \sqrt{(l)/(g)}}\\\\\therefore \omega =\sqrt{(g)/(l)}`

As we can see that the angular frequency is independent on the initial angle (valid strictly for small angle approximations) we conclude that the angular frequencies of the 3 pendulums are the same.