what is moment of inertia in engineering mechanics?

Knowing how the formula for Moment of Inertia is derived may help you understand why it takes the form it does and why it is correct. It will also relate to other (linear) formulas that you know. Unlocking the secrets of formulas you use will take you far beyond those who blindly apply them. Good luck.

Understanding WHY moment of inertia is decribed as I=mr^2 is more of a physics question than an engineering question.

For simplicity, let us consider a point particle with mass m rotating about a given axis. The system can be generalized as the sum over all point particles to understand the system. This particle rotates about the axis at a radius of r with a frequency w. It is known that the rotational kinetic energy of this particle is

T(rot) = .5*m*(w x r)^2

[for w x r as vector cross product]. By a mathematical identity, this is

T(rot) = .5*m*{w^2*r^2 - (w . r)^2}

[where w . r is the dot product]. This can be expressed as in component form to give you a 3 by 3 matrix for I [in 3 dimensional space] called the inertia tensor.

T(rot) = .5 * sum(ij)[ I(ij) * w(i) * w(j) ]

I(ij) = m * { delta(ij)sum(k)[x(k)^2] - x(i)*x(j) }

[where delta(ij) is the Kronicker delta: is 1 when i=j, else 0]

For an elementary treatment when the particle is always only moving in a direction that is perpendicular to the axis of rotation, this inertia tensor will only have diagonal elements and a scalar I is described where I = sum(ij)[I(ij)] which is the moment of inertia you are familiar with. Now,

T(rot)=.5 * I * w^2

I = m * r^2

Essentially, the square of the length term comes from the fact that a rotating particle takes more energy to rotate it based on how far away it is from the axis as well as how quickly it tries to rotate (which is also proportional to the radius).

moment of inertia is taken for rigid bodies about a center

to consider the whole body we consider

its revolution about the axis about which inertia is to be calculated

and this revolution of the elementary part of the body leads to insertion of term x square