What is moment of inertia?

Question

Folks, my engineering structures teacher told me moment of inertia is nothing but resistance to bending.
Though I understand it, I cant ..... well its a hard feeling to explain.

1)what is moment of inertia?
2)What is the formula to calculate it for various shapes?
3)how do we get the formula?
4)How to find the moment of inertia of a square of side 1 m?

Please help me out.

Answer 1

Moment of inertia can be thourght of as resistance to ROTATION. Where it comes from is altering Newton's laws to deal not with spatial displacements, but angular displacements. This is convenient for rigid bodies that rotate about an axis. I'll give an example in a bit.

First of all, in physics, we always use radians as our angle measure. This is because theta=s/r where s is an arc-length of a circle subtended by the angle theta. What this also means is that if the angle is changing, the rate of change of arc length--the actual speed of something undergoing the rotation --is proportional to the time rate of change of angle TIMES R. Think of a spinning CD--the outside spins faster than the inside . . . The easiest example of course is something where everything is roughly at the same radius, like a ball moving in a circle or a spinning hoop (like a wheel). Now if we use torque, Fxr, which is r*(Ftangential) in magnitude, and we know v(tangential) =r(omega), where omega is the ANGULAR SPEED, then a(tangential) = r(alpha), where alpha is the ANGULAR acceleration. So . . .
if we say Torque = r*Ft = I(alpha) then if I = mr^2 we get:

rF=mr(r(alpha))

or

F(tan)=ma(tan) Newton's second law.

In order to get the moment of inertia for a larger body, you must sum these little r^2(dm) guys all over--and the axis matters. In other words, say if z is my axis of rotation, I need to do the integral:

r^2dm = (x^2+y^2)dm=(x^2 + y^2)(density)dV =

(x^2+ y^2)(density)dxdydz

Over the geometry of the body. Cartesian coordinates, of course, are not always the brightest way to do this.

In order to answer 4) we need to know where the cube is going to be spinning about. It's moment of inertia will be at a minimum if it is upright and spinning about the center.