# How to increase second moment

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## Main Question or Discussion Point

Hi there,
I am reading a math book with a lot of examples on mechanical physics. I saw a math term about second moment. In wiki, it is said that second moment is just moment of inertia in physics and has definition as below

$\int(x-\mu)^2f(x)dx$

here $\mu$ is the average and $f(x)$ is the weight or probability. Let me rewrite this into the summation with uniform weight as follow

$\dfrac{\sum_i (x_i-\mu)^2}{N} = \langle x^2\rangle - \langle x\rangle^2$

I am trying to associate this with moment of inertia and try to figure out how can we increase the moment of inertia based on this formula. In that formula, if we want to increase the second moment, we should decrease the average of $x$, if $x$ is mass, does it mean that I need to put all mass pieces as close to others as possible so to have minimum average? If I did that, how can I tell that won't decrease the first term as well? Thanks.

mathman
Your best bet is to increase $\langle x^2\rangle$ while keeping the mean constant. This means moving things away from the mean on opposite sides.

andrewkirk