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Moment of inertia for rectangular plate

by siestrand
Tags: inertia, moment, plate, rectangular
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siestrand
#1
Apr17-11, 03:08 PM
P: 3
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.
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#2
Apr17-11, 08:26 PM
P: 106
Consider the plate as made of several paralel slices.
siestrand
#3
Apr18-11, 06:53 AM
P: 3
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
#4
Apr18-11, 07:35 AM
P: 1
Moment of inertia for rectangular plate

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
siestrand
#5
Apr18-11, 07:50 AM
P: 3
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.


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