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 [tex] \frac{1}{12}ml^2 [/tex]. 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.
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?
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 havent used any integrals..its just addition
There is one mistake, I think. One slice has of course [tex] I = m(a^2)/(12n) [/tex] but not by this axis! the axis is in the centre of plate and every slice has [tex] I = m(a^2)/(12n) + \frac{m}{n} * r^2 [/tex] from Steiner theory when r ist distance from axis.