Moment of Inertia about the line x=y

  • Thread starter Jadehaan
  • Start date
  • #1
24
0

Homework Statement


Let [tex]\Omega[/tex] represent a two dimensional material region who density is given by [tex]\rho[/tex](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: [tex]\int[/tex][tex]\int[/tex]x2[tex]\rho[/tex](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.
 

Answers and Replies

  • #2
Dick
Science Advisor
Homework Helper
26,260
619
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?
 
  • #3
24
0
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.
 
Last edited:
  • #4
Dick
Science Advisor
Homework Helper
26,260
619
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.
 
  • #5
24
0
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?
 
  • #6
Dick
Science Advisor
Homework Helper
26,260
619
Yes, that sounds correct. So r^2=(x-y)^2/2.
 
  • #7
24
0
Yep, thanks so much Dick.
 

Related Threads on Moment of Inertia about the line x=y

  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
0
Views
2K
  • Last Post
Replies
12
Views
7K
Replies
1
Views
2K
Replies
29
Views
3K
  • Last Post
Replies
9
Views
4K
Replies
3
Views
2K
  • Last Post
Replies
4
Views
1K
  • Last Post
Replies
1
Views
823
Top