Moments of Inertia by Integration

[CivilSigma]

Homework Statement

I am having trouble understanding the formula for Moments Of Inertia by direct integration.

Homework Equations

I understand the following (which is the definition) :

$$ I_x = \int y^2 dA $$ $$I_y = \int x^2 dA $$

However come to application on a problem, my book doesn't even use those formulas.

The authors derived a new formula:

$$ dI_x =\frac{1}{3} y^3 dx $$ $$I_x = \int dI_x$$

Here is a screen shot of what they did:

http://tinypic.com/r/2vjr5vq/8

Why do they do that? Why not just use the definition and solve?

The Attempt at a Solution

 

[andrewkirk]

The first formula for $I_x$ is a summation over horizontal strips between $y$ and $y+dy$. The integration is over $y$ so in the integrand $y$ is the integration variable. The strips are not assumed to be all the same width, so the shape being integrated need not be a rectangle, and need not touch the $x$ axis.

The second formula for $I_x$ is a summation over vertical strips between $x$ and $x+dx$. The integration is over $x$. In the integrand $y$ represents the height of the top of the vertical strip above the $x$ axis. It assumes that the base of the strip is the $x$ axis.

The second formula is derived from the first, by using the formula for MOI of a vertical rectangle that was just derived.

The alternative formulation is provided because sometimes it will be easier to integrate over $x$ than over $y$. In any given situation, one can choose the method that gives the easiest calculation. Note however that the first method is more general, as the second can only be used when the base of the shape is flat and runs along the $x$ axis, and the shape has no parts that project sideways beyond the base.

 

[CivilSigma]

Thank you for the reply andrewkirk!

I have to understand the use of $dI_x$ in the formula because we are summing up the moments of inertia of all the rectangles.

So, I'm just curious, how would I go about solving problems given the original (definition) equation for moments inertia?

 

[andrewkirk]

how would I go about solving problems given the original (definition) equation for moments inertia?

You use the first formula you've written above, and you replace $dA$ by $w(y)dy$ where $w(y)$ is the width of the shape at height $y$.

 

[CivilSigma]

For both $I_x$ and $I_y$ ?

 

[andrewkirk]

No, just for $I_x$. To do $I_y$ you swap $x$ and $y$ in all formulas.

By the way, the code to get in-line latex math symbols on physicsforums is two consecutive # characters, not a single $. That's an annoying difference with Texmaker, which is what I use elsewhere.

 

[CivilSigma]

Thank you Andre! I'll try some problems and hopefully I get them right.