Relation between angular velocity and linear velocity in a pendulum

Question

For a pendulum in simple harmonic motion, I understand that the period is independent of amplitude, and I know that the angular velocity is inversely proportional to the period. However, doesn't the angular velocity change as linear speed changes?

Answer 1

Yes there is a very simple relationship between the angular velocity and the tangential velocity, given by $v_t = \omega r$, where $v_t$ is the tangential velocity, $\omega$ is the angular velocity and $r$ is the radius of the pendulum.

Answer 2

There are two different concepts of angular velocity associated with a pendulum, and you have mixed up the two.

The displacement of the pendulum is given by something like:

$$ x = A \sin(\omega t) \tag{1} $$

where $\omega$ is a constant called the angular velocity and related to the frequency and period by:

$$ \omega = 2\pi f = \frac{2\pi}{\tau} $$

However there also the angluar velocity of the pendulum bob around the pivot point which is given by:

$$ \Omega = \frac{v}{r} $$

and if we differentiate equation (1) to get the velocity we get:

$$ \Omega = \frac{A\omega}{r} \cos(\omega t) \tag{2} $$

so this second type of angular velocity is not a constant. Your confusion arises because you have mixed up the constant $\omega$ and the variable $\Omega$.