The question is..?

Question

1) A stick of length L = 2.0 m oscillates as a physical pendulum. (a) What value of distance x between the stick's center of mass and its pivot point O gives the least period? (b) What is that least period?

Answer 1

The period of the pendulum is

T = 2 pi sqrt ( I / mgd )

where I is the moment of inertia of the pendulum about the pivot point and d is the distance between the pivot point and the center of mass.

You want the period to be a minimum. That means you want to increase d. However, this also increases I, which increases the period. However, it turns out that increasing d is your best bet. If you hang a rod of mass M and length L rod by its very end, the moment of inertia is

I = 1/3 M L^2

In this configuration the parameter d = 1/2 L

Plug all of this into the first equation and solve for T.

Plug those values into the first equation and solve for the period T.