Parallel Axis Theorem Clarification

[matt57017]

in what situations would you require the use of the parallel axis theorem?

Also, from the physics book it says that let x and y coordinates of P(a point parallel to the first axis) be a and b. then let dm be a mass element(what does this mean? a point anywhere within the object?) with the general coordinates x and y. the rotational inertia of the body about the axis through P is then I=∫r^2 dm = ∫[(x-a)^2 + (y-b)^2]dm

im a little confused and any clarification would as to how, why, and when this would make sense.

thanks!

 

[Infrared]

Moment of Inertia, I is defined as $\int {r^2}\,dm$ where dm takes on the value of the masses of all infinitesimal pieces of area. This makes sense since for a finite collection of masses, $m_i$, moment of inertia is defined as
$I= \sum_{i} r_i^2 m_i$.

The parallel axis theorem, which states that $I = I_{cm} + mr^2$ is useful when a mass is being rotated about an axis which does not go through the center of mass. For example, if I am rotating a uniform disk of mass M and radius R about an axis perpendicular to the disk and a distance of r away from its center, then the moment of inertia for this rotation would be $I= {\frac{MR^2}{2}}\ + Mr^2$.