# Thread: center of mass with varying density

1. ## center of mass with varying density

Hi, I'm having incredible trouble getting a reasonable answer to this problem:
Find the center of mass of a thin plate covering the region bounded below by the parabola y=x^2 and above by the line y=x if the plate's density at the point (x,y) is d(x)=12x. I have an idea of what to do for the x coordinate, but I'm confused as to how to do the y coordinate. Thanks!

2. Originally Posted by musicman314
Hi, I'm having incredible trouble getting a reasonable answer to this problem:
Find the center of mass of a thin plate covering the region bounded below by the parabola y=x^2 and above by the line y=x if the plate's density at the point (x,y) is d(x)=12x. I have an idea of what to do for the x coordinate, but I'm confused as to how to do the y coordinate. Thanks!
We can also write d(x,y) = 12x. Does that make it look more doable?

I think you'll want to integrate with respect to y first, because then the innermost integrand will just be a constant (with respect to y). The limits on your integral should be

outer: 0 to 1

inner: x^2 to x

EDIT: I was thinking in a probability distribution mode. I'll have to see how well it applies to center of mass. Anyway the integral limits should be fine.

3. A quick question, is that a double integral that you were describing? I probably could do that, but since this is for a calc 1 class, the professor might not appreciate that too much...

4. Originally Posted by musicman314
A quick question, is that a double integral that you were describing? I probably could do that, but since this is for a calc 1 class, the professor might not appreciate that too much...
All right, yes, sorry that was a double integral I was describing, and turns out I was telling you how to calculate the mass, not the center of mass.

So of course you're going to calculate the x-coordinate and the y-coordinate separately... I think this thread could help you, I'm currently looking it over.

Edit: Oy, that link I gave you is for a homogeneous plate. Sorry again.

5. Originally Posted by musicman314
Hi, I'm having incredible trouble getting a reasonable answer to this problem:
Find the center of mass of a thin plate covering the region bounded below by the parabola y=x^2 and above by the line y=x if the plate's density at the point (x,y) is d(x)=12x. I have an idea of what to do for the x coordinate, but I'm confused as to how to do the y coordinate. Thanks!
$\displaystyle C.O.M = (\bar x, \bar y)$

$\displaystyle \bar x = \frac{ \iint x \rho dxdy } { \iint \rho \space dxdy }$

$\displaystyle \bar y = \frac{ \iint y \rho dxdy } { \iint \rho dxdy }$

But since your density only depends on X, those double integrals can be reduced to single integrals. This can also be done by parametrization of x and y.

6. Originally Posted by AllanCuz
$\displaystyle C.O.M = (\bar x, \bar y)$

$\displaystyle \bar x = \frac{ \iint x \rho dxdy } { \iint \rho \space dxdy }$

$\displaystyle \bar y = \frac{ \iint y \rho dxdy } { \iint \rho dxdy }$

But since your density only depends on X, those double integrals can be reduced to single integrals. This can also be done by parametrization of x and y.
OK, I'm pretty sure this method is valid, avoiding double integrals.

First we need mass M.

$\displaystyle M = \int_a^b \rho(x) (g(x)-f(x)) dx$

Where in this case $\displaystyle g(x) = x$, $\displaystyle f(x) = x^2$, $\displaystyle \rho(x)=12x$, $\displaystyle a = 0$ and $\displaystyle b = 1$.

Then we write

$\displaystyle \bar x = \frac{1}{M} \int_0^1 x\rho(x)(g(x)-f(x)) dx$

$\displaystyle \bar y = \frac{1}{M} \int_0^1 \rho(x)\left(\frac{1}{2}\right)(g^2(x)-f^2(x)) dx$

I got these equations by combining this reference (which seems to have a typo) and this reference, with modifications.

7. Originally Posted by undefined
OK, I'm pretty sure this method is valid, avoiding double integrals.

First we need mass M.

$\displaystyle M = \int_a^b \rho(x) (g(x)-f(x)) dx$

Where in this case $\displaystyle g(x) = x$, $\displaystyle f(x) = x^2$, $\displaystyle \rho(x)=12x$, $\displaystyle a = 0$ and $\displaystyle b = 1$.

Then we write

$\displaystyle \bar x = \frac{1}{M} \int_0^1 x\rho(x)(g(x)-f(x)) dx$

$\displaystyle \bar y = \frac{1}{M} \int_0^1 \rho(x)\left(\frac{1}{2}\right)(g^2(x)-f^2(x)) dx$

I got these equations by combining this reference (which seems to have a typo) and this reference, with modifications.
This is correct. You would get to this point had you evaluated the double integrals for dy only (leaving dx as a single integral).

8. Originally Posted by AllanCuz
This is correct. You would get to this point had you evaluated the double integrals for dy only (leaving dx as a single integral).
Yeah, assigning this problem in a non-multivariable calculus class is like a lot of introductory physics classes -- give people the equations without saying where they came from.

9. thanks much guys! I have no idea why we're even doing this for a calc 1 class, since my friends started with stuff like this in calc 2. At least when I get outta high school, I'll have a teacher for my math class. (UW-Extension: not my best choice)