- #1
ConnorM
- 79
- 1
Homework Statement
Hello PF! I'm having some trouble on the last part of my assignment, it's question 4 part "c".
Here is a picture of the assignment [http://imgur.com/1edJ3g5] ! I'll post this instead of writing it out so we know that we're all looking at the same thing!
Homework Equations
The change of variables given at the beginning of question 4 are,
x=au, y=bv, z=cw
From part "a" I used the change of variables given in the question and found that the ellipsoid equation became u^2 + v^2 + w^2 = 1. I found the Jacobian to be equal to abc. Next I set up my integral to determine the volume over the region S, ∫∫∫abc dV, Since a sphere with the radius 1 will have a volume of 4pi/3 I found my volume to be abc*4pi/3.
I think what I need for part "c" is just the Jacobian. so the Jacobian = abc.
The equation for inertia that we were given in class was I=∫∫(x^2 + y^2)*ρ(x,y) dA
Changing from rectangular to spherical coordinates. (I think you need to use this)*
x = ρsin(β)cosΘ
y=ρsin(β)sinΘ
z=ρcos(β)
The Attempt at a Solution
So to start off since I'm working in 3 Dimensions would I have to change my formula for moment of inertia to,
I=∫∫∫(x^2 + y^2 + z^2)*ρ(x,y,z) dV,
Then from here since I am working with changed variables I changed the x, y, and z, also multiplied by the Jacobian,
I=∫∫∫((au)^2 + (bv)^2 + (cw)^2)*ρ(x,y,z)*abc dV
From here would I have to switch to spherical coordinates? I would obtain,
I=∫∫∫((ρsin(β)cosΘ)^2 + (ρsin(β)sinΘ)^2 + (ρcos(β))^2)*ρ(ρ,Θ,β)*abc dV
I=∫∫∫(ρ^2)*ρ(ρ,Θ,β)*abc*(ρ^2)sin(β) dρdΘdβ
Then my bounds of integration would be
0 ≤ ρ ≤ 1
0 ≤ Θ ≤ 2pi
0 ≤ β ≤ pi
Does this look right so far, Or am I off track? if it looks good just let me know and I will continue, I'll reply as soon as I have either finished or got stumped again!