Moment of inertia for rectangular plate

[siestrand]

Hi!
I've got a problem with this:
Count moment of inertia for rectangular plate a x b, if you know that moment of inertia of thin rod is $$\frac{1}{12}ml^2$$. Do not use integrals!, others mathematical functions required (I can proof this moment by integrals, but this is not issue). I know that I have to use Steiner theory, but how? No integrals? :(
Please help.

 

[Kittel Knight]

Consider the plate as made of several parallel slices.

 

[siestrand]

Ok, I know that, but the problem is: how can I count it without integrals? I must sum up all the slices' distance from axis, but how am I supposed to do it without integrals?

 

[sharmaphy]

let us consider a rectangular plate to be a x b dimensions of total mass m

let x-axis be along the length direction and y be along width direction and origin be at the center of plate.
consider it to have 'n' parallel slices (n being very large number) along x direction so that each slice a like a rod of mass m/n and length 'a'
Moment of inertia of each rod along x axis, I = m(a^2)/(12n)
Moment of inertia of plate along x aixs =Ix = n I = m(a^2)/(12)
( this is because u have n slices)
similary if u repeat above exercise along y direction
Moment of inertia of plate along y aixs = Iy = m(b^2)/(12)

Moment of inertia of plate along z aixs = Iz = Ix + Iy
= m(a^2)/(12) + m(b^2)/(12)
= m( a^2 + b^2)/12

note : we haven't used any integrals..its just addition

 

[siestrand]

There is one mistake, I think.
One slice has of course $$I = m(a^2)/(12n)$$ but not by this axis! the axis is in the centre of plate and every slice has $$I = m(a^2)/(12n) + \frac{m}{n} * r^2$$ from Steiner theory when r ist distance from axis.