# Moment of inertia of a few basic objects

Draco27
Ok i need some help with some homework that is to derive formula for moment of inertia of a few objects about the axis's that i have mentioned
1. Rectangular slab about axis through center(sides a,b)

The only equation i know is Moment of inertia = ∫r$^2$dm

any help would be appriciated

I am new here so if this has been answered pls help me locate and lock this up...

Muphrid
Where are you getting stuck? You don't know how to apply the formula you were given? Something else?

Draco27
Hmm i could do it for sphere or ring or others but got struck on slab

But i could not understand the double integration or how it is done...

So pls help me solve that if possible...

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Muphrid
How did you set up the double integral?

voko
What does "axis through center" really mean? There are infinitely many possible axes through the center of a figure.

Draco27
To Muphrid

Ok What i did is this

rho=M/a*b

dm=rho*dx*dy (assuming smaller rectangles of length dx and dy)

di= dI = r2 * rho* da*db

dI = (r^2 * m * da* db )/ab

Now it says to integrate with limits from -a/2 to a/2 and -b/2 to b/2

also couldnt understand what to put in r

To voko

the axis is passing through center and perpendicular to plane of rectangle

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voko
r is the distance from the axis to the (x, y) point.

Draco27
But this distance changes right so what exactly i put??

Also if possible help me reach a solution
i mean what would u do to solve??

voko
The axis is at the point (0, 0). What is the distance between that point and (x, y)?

Muphrid
To Muphrid

Ok What i did is this

rho=M/a*b

dm=rho*dx*dy (assuming smaller rectangles of length dx and dy)

di= dI = r2 * rho* da*db

dI = (r^2 * m * da* db )/ab

Now it says to integrate with limits from -a/2 to a/2 and -b/2 to b/2

also couldnt understand what to put in r

Don't call them $da, db$. They're $dx$ and $dy$. This is what you have:

$$\int_{-b/2}^{b/2} \int_{-a/2}^{a/2} \rho r^2 \; dx \; dy$$

You're confused about what to put in for $r$. It should represent the distance to the axis of rotation, but you're integrating in terms of $x,y$. Is there some way you can put $r$ in terms of $x,y$?

voko
You are not following the advice given to you. You need to express r in terms of x and y.

Draco27
Thats the thing i cant figure out

Do u have a solution??

Muphrid
Have you heard of the Pythagorean theorem?

Draco27
But the r from the central axis changes all around the rectangle.....

voko
Are you saying that you are tasked to compute moments of inertia without studying the basic properties of the Cartesian coordinate system? I fail to see the point of such an assignment.

Muphrid
But the r from the central axis changes all around the rectangle.....

You're not being asked for the distance all around the rectangle, but all throughout the rectangle. Yes, this distance changes as you move within the slab. That's fine. What's important is you have an expression in terms of $x,y$. Once you do that, you can integrate it.

You should not expect the distance to be a constant.

Draco27
So which distance do i put in the place of r??

Muphrid
What do you mean which distance? Is there more than one you think might be correct?

Draco27
so i put r2=x2+y2

after that??

how i solve the double integration??

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voko
Do you know how to take a single integral?

Draco27
of course

Muphrid
Integrals are linear, which means you can do the integrals for $x^2$ and $y^2$ separate and then just add them together at the end.

voko
Then take the single integral by x first. Treat y as a constant. Then take the integral by y.

Draco27
and which limit do i put after integrating??

there are 4 limits.....

would appreciate if u could solve......

voko
Choose a-limits for x, and b-limits for y. Or the other way around. It does not change anything.

Draco27
Finally...
Got it
Thanks man
Many many thanks...to u and Muphrid