# Calculating Rotational Inertia of Flat Plate

• azizlwl
In summary, to derive the rotational inertia for a flat plate, you can use the formula I = ∫r²ρdV, where r represents the distance from the point being evaluated to the axis of rotation and ρ is a function of r. This integral must be evaluated in two dimensions for a flat plate, as opposed to one dimension for a thin rod. Using the parallel axis theorem, you can calculate the moment of inertia for a small rectangular element and integrate to get the final answer.
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.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.

## 1. What is rotational inertia and why is it important?

Rotational inertia, also known as moment of inertia, is a measure of an object's resistance to changes in its rotational motion. It is important because it determines how much torque is needed to accelerate an object in rotational motion.

## 2. How do you calculate the rotational inertia of a flat plate?

The formula for calculating rotational inertia of a flat plate is I = (1/12) x m x (a^2 + b^2), where I is the rotational inertia, m is the mass of the plate, and a and b are the dimensions of the plate.

## 3. What are the units of rotational inertia?

The units of rotational inertia are kg x m^2. This can also be written as kg x m^2/s^2.

## 4. Can the rotational inertia of a flat plate change?

Yes, the rotational inertia of a flat plate can change if the mass or dimensions of the plate change. It can also change if the plate's axis of rotation changes.

## 5. How is rotational inertia different from mass?

Rotational inertia and mass are two different measurements. Mass measures the amount of matter in an object, while rotational inertia measures an object's resistance to changes in rotational motion. An object can have a large mass but a small rotational inertia, or vice versa.

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