Can someone please explain to me once and for all, why is the moment of inertia of a body $A$ Is calculated as:
$$I_x = \int_A y^2 dA ,\quad I_y= \int_A x^2 dA .$$
I searched a lot google for a summary and derivation, but couldn't find any good one that explains in detail the derivation of this formula.
Do you have a reference for this fact?
As for the center of gravity , (using double integrals) do you have a good reference for it?
Thanks !!
Best Answer
http://en.m.wikipedia.org/wiki/Moment_of_inertia the section where it mentions calculating the moment of inertia give a short derivation based on energy concerns why the moment of inertia is given by $$ \sum mr^2 $$. This is a discrete sum which is then given a continuous definition in terms of integration. For example an object rotating about the y axis ( if you remember your cylindrical shells method) has a radial distance x, so to sum up terms of the form $$ mx^2 $$, we treat each tiny volume element as a term $$ x^2 dm = x^2 \rho dA $$ summing up yields $$\int \int x^2 dA$$ for an object of constant unit density. Interchange the x and y axis for the other result.