# Moment of Inertia about the line x=y

In summary, the moment of inertia of a two dimensional material region with density given by \rho(x,y) about the line y=x is given by the integral of (x-y)^2/2 \rho(x,y) dA, where r is the distance from the point (x,y) to the line x=y. This can be found using the formula r=|x-y|/ \sqrt{2} and substituting it into the moment of inertia formula.

## Homework Statement

Let $$\Omega$$ represent a two dimensional material region who density is given by $$\rho$$(x,y). Establish an integral formula for the moment of inertia of the material region about the line y=x.

## Homework Equations

The moment of inertia about the x axis: $$\int$$$$\int$$x2$$\rho$$(x,y)dA

## The Attempt at a Solution

I am attempting to transform the moment of inertia about the x-axis to the line x=y, how would I go about doing this?

Thanks for any help, Jim.

The moment of inertia is the integral of r^2 rho(x,y) dA where r is the distance from the point (x,y) to the axis. That means the expression you gave is the moment of inertia about the y axis, not the x axis. Now you have to replace r in the integral with the distance from a point (x,y) to the line x=y. How many ways do you know to find the distance from a point to a line?

Thanks for your help. You were right, I mistyped the formula for the moment about the x axis. Using points (x,y) and (y,x) I calculated r2=2x2-4xy+2y2 So the formula would be the integral of this multiplied by rho dA.

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That seems close, but aren't you off by a factor of four there? The distance from (1,0) to the line x=y is sqrt(2)/2. Isn't it? So that makes r^2 for (1,0) r^2=1/2.

I see. I was using the wrong distance formula. So using the formula for the distance between a point (m,n) and a line Ax+By+c=0 the distance is r= |Am+Bn+C|/ sqrt(A2+B2)
Using the point (x,y) and the line x-y=0 I calculated the distance r=|x-y|/sqrt(2)
Does this sound correct?

Yes, that sounds correct. So r^2=(x-y)^2/2.

Yep, thanks so much Dick.

## What is moment of inertia about the line x=y?

Moment of inertia about the line x=y is a measure of an object's resistance to changes in rotational motion around the line x=y. It takes into account the mass distribution of the object and its distance from the line x=y.

## How is moment of inertia about the line x=y calculated?

Moment of inertia about the line x=y is calculated by integrating the squared distance from the line x=y for each infinitesimally small element of mass in the object, multiplied by the mass of that element. The resulting value is summed up for all elements of mass in the object.

## What is the significance of moment of inertia about the line x=y?

The moment of inertia about the line x=y is important in understanding the object's rotational behavior. It affects the object's stability, the amount of torque required to rotate it, and the object's angular acceleration.

## How does the moment of inertia about the line x=y differ from the moment of inertia about other axes?

The moment of inertia about the line x=y is specific to the object's rotation around that particular axis. It is different from the moment of inertia about other axes because it takes into account the object's distribution of mass relative to that specific axis.

## Can the moment of inertia about the line x=y be changed?

Yes, the moment of inertia about the line x=y can be changed by altering the object's mass distribution or its distance from the line x=y. This can be done by changing the shape, size, or placement of the object's mass.

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