Calculating Rotational Inertia of Flat Plate

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The discussion focuses on deriving the rotational inertia of a flat plate, emphasizing the need to evaluate the integral in two dimensions rather than one, as with a thin rod. The formula for rotational inertia is presented as I = ∫ r² ρ dV, where r is the distance from the axis of rotation and ρ may depend on r. Participants clarify that the integral should be evaluated over the volume of the body, and they suggest using a small rectangular element to apply the parallel axis theorem for calculations. The conversation highlights the importance of understanding the geometry involved in the integration process. Overall, the thread provides insights into the mathematical approach for calculating rotational inertia for flat plates.
azizlwl
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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.http://imageshack.us/a/img845/1134/53433231.jpg
 
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Well, you just calculate it from the definition:

I = \int r^{2} \rho \mathrm{d}V

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.
 
Thank you.
Can you elaborate more.
Can you tell me what r represents in your equation?
Your equation is antiderivatives not definite integral.
 
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Okay, r is the distance of the point you're evaluating to the axis of revolution. So \rho is generally a function of r too.

It is a definite integral (I should have written \int_V instead of \int. 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.
 
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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