Here is a question based on Simple Harmonic Motion that I tackled just now. However I think I am having an approach to tackle this but I am not sure about it.
Ouestion: A uniform disc of radius $R$ is pivoted about point $P$ such that it is free to oscillate in the vertical plane. Distance between the pivot and centre of disc $x$, such that the time period of oscillation is minimum.
So till yet I know that for a circular disc like this one, we have time period, $$T=2\pi\sqrt{\frac{I}{\kappa}}$$ where $I$ is moment of inertia of the disc and $\kappa$ is the torsional constant.
So after substituting the values knowing that $\kappa=\frac{\tau}{\theta}$ and $I=mr^2$ I have,
$$T=2\pi\sqrt{\frac{mr^2\theta}{\tau}}$$
Now next to this I am thinking of finding the minima of this function so that I can get the least value of r. However I am still not sure that this will help me in anyway.
Also I want to know that if we can differentiate this function to get minimum value of $r$ then do we really need to substitute $\tau$ as $\tau=r\cdot F$.