Thread: Multi - Balancing An Object

Suppose we have a landmass of an island that needs to be balanced by an elephant using its tusk. We need to find the right place to place the elephant's tusk to balance it. The center of the universe is some sdistance away from this island (in the same plane as this island). If we measure from the center of this universe, the island is bounded by the following functions:

y = x^2 + 4 and y = x + 6

If the density of the island at point (x,y) is the inverse of the square (reciprocal) of the distance from the x-axis, where should the tusk be placed in order to balance the island?

Look below. I will not solve, I will set up.

Originally Posted by ThePerfectHacker

Look below. I will not solve, I will set up.

Hello, and thanks in advance for your time

I'm familiar with the formula

int(int_R f(x,y)))dA which is what you have but "D" .. same thing.

Now, I don't know why you have an extra "x" and "y"..only time I've seen this is when finding the volume using polar coordinates, where the dA then becomes r*dr*d(theta) .. a variation of fubini's theorem.

Then, I see how you got the region by:

x^2 + 4 <= y <= x + 6.. but don't see how you got the -2 <= x <= 3 ..

And further, I have no idea what (delta, I think it is)*x = k (curvature?)/x^2 is ... and then you have:

x_bar = M_y/M and y_bar = M_x/M (this reminds me of the chinese remainder thm, but obviously this isn't number thry!). I suspect it's "mass" . This part got me lost.. the final step in your work is the one that kind of makes sense:

M = int(int_R f(x,y)))dA, but what is our f(x,y) .. and limits of integration? That'd be the region you described? We'd use fubini's theorem to compute this, such that:

int (from c to d) of int (from a to b) of f(x,y) dy dx

OR, obviously, int (from a to b) of int (from c to do) of f(x,y) dx dy

Thanks for the help, and I apologize for not understanding your work.

EDIT: Ohh!! I kind of see now how you got the M_x, M_y..

I remember this being tricky..

M_x = mass of lamina ... the x-coordinate of the center of mass.. = M_y/m .. and then similarly for M_y = M_x/m..

Now it makes sense why you had the extra x, y.. but I still don't get what limits correspond to which integral and what f(x,y) is.

Take 2:

So now that it makes a little more sense to me, would we have the following:

M_x/m (where m is the total mass, and M_x is the moment with respect to the x-axis) = int*int_R y*f(x,y) dA

And then for M_y/m = int*int_R x*f(x,y) dA

But first we want to find what m is, which I believe is just

int*int_R f(x,y) dA ..

So this should be simple enough I think after I know what f(x,y) is.. and how you got the limits of integration.

Going back to the word prob., what exactly does it mean by:

"If the density of the island at point (x,y) is the inverse of the square (reciprocal) of the distance from the x-axis..." isn't that just f(x,y)? Is that how you figure out f(x,y)?

Originally Posted by Ideasman

x^2 + 4 <= y <= x + 6.. but don't see how you got the -2 <= x <= 3 ..

I think I might have made a mistake.

Ahh okay, so it is -1 <= x <= 2. Thanks for confirming it !

So now the only barrier stopping me from doing the various integrals and finding M_y/M and M_x/M is finding what f(x,y) is .. you have the delta*x = k/x^2 and I have no idea what that is.. (I hope your "k" doesn't mean curvature, or then you totally lost me).

that "inverse of the square" wording is very confusing, and I have no idea how to come up with a function based on that information. Once I find this out the rest will follow from there, since I will have the func, and then finding x_bar and y_bar are easy, along with mass which is all the info. I need for the problem.

Originally Posted by Ideasman

Ahh okay, so it is -1 <= x <= 2. Thanks for confirming it !

So now the only barrier stopping me from doing the various integrals and finding M_y/M and M_x/M is finding what f(x,y) is .. you have the delta*x = k/x^2 and I have no idea what that is.. (I hope your "k" doesn't mean curvature, or then you totally lost me).

"k" is just a constant function (positive). Like 2 or 3 or pi.
Just treat it like a number. In the end it shall cancel.

that "inverse of the square" wording is very confusing, and I have no idea how to come up with a function based on that information. Once I find this out the rest will follow from there, since I will have the func, and then finding x_bar and y_bar are easy, along with mass which is all the info. I need for the problem.

Inverse - as in "inversely proportional".
Thus, an inversely proportional square is 1/x^2

But you need to include "k" in front because there is also something called: the konstant of proportion.