Rotational Inertias.

azizlwl

How to derive the Rotational Inertia for a flat plate.
I do not see any symmetry about the axis of rotation as in a thin rod.

jahaan

Well, you just calculate it from the definition:

[tex]I = \int r^{2} \rho \mathrm{d}V[/tex]

The difference here is that the integral has to be evaluated in two dimensions. For a thin rod you can eliminate two coordinates (it's a one-dimensional body); this is a two-dimensional body.

azizlwl

Thank you.
Can you elaborate more.
Can you tell me what r represents in your equation?
Your equation is antiderivatives not definite integral.

jahaan

Okay, r is the distance of the point you're evaluating to the axis of revolution. So [tex]\rho[/tex] is generally a function of r too.

It is a definite integral (I should have written [tex]\int_V[/tex] instead of [tex]\int[/tex]. You integrate over the volume of the body. That's why you only evaluate in one dimension for a thin rod, but two dimensions for a thin plane.

ashishsinghal

Hey Azizlwl,

Take a rectangular element of length a and height as given at a distance r from the axis. Make the width very small i.e. dr. What is the moment of inertia of this element about the axis? (Use parallel axis theorem). Now integrate. you will get your answer.

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azizlwl	Rotational Inertias. Tweet How to derive the Rotational Inertia for a flat plate. I do not see any symmetry about the axis of rotation as in a thin rod.

jahaan	Well, you just calculate it from the definition: [tex]I = \int r^{2} \rho \mathrm{d}V[/tex] The difference here is that the integral has to be evaluated in two dimensions. For a thin rod you can eliminate two coordinates (it's a one-dimensional body); this is a two-dimensional body.

jahaan	Rotational Inertias. Okay, r is the distance of the point you're evaluating to the axis of revolution. So [tex]\rho[/tex] is generally a function of r too. It is a definite integral (I should have written [tex]\int_V[/tex] instead of [tex]\int[/tex]. You integrate over the volume of the body. That's why you only evaluate in one dimension for a thin rod, but two dimensions for a thin plane.

ashishsinghal	Hey Azizlwl, Take a rectangular element of length a and height as given at a distance r from the axis. Make the width very small i.e. dr. What is the moment of inertia of this element about the axis? (Use parallel axis theorem). Now integrate. you will get your answer.