- #1
dlivingston
- 16
- 0
Calculus 3 project – any and all help is appreciated.
We've just gone over center of mass with double integrals, so it's a bit peculiar to see this project feature only one integral. I went over that in calc 2 – and as a result, know how to calculate it that way.
However (as you will see in the formula below), I don't think he wants me to calculate it that way. I have no idea how to proceed and the book is of absolutely no help.
Anything you could do to help me out would be fantastic.
And here's the problem:
There are three type of laminae we are trying to find:
1) A triangle with sides 3", 4", and 5" respectively.
2) A semicircle
3) A "horseshoe", i.e. a half-ring, whose inner and outer edges are composed of circular arcs of radii 3" and 5" respectively.
Definition 1:
The first moment of a planar mass distribution about a line l is the integral over the region of the (area) density σ(P) times the distance of P from the line. For example, the first moment about the line x=k is
[itex]M_k=∫_R\left(x-k\right) σ\left(P\right) dA[/itex]
Definition 2:
In order for a line to be a balancing line, the first moment about this line must be zero.
See the M_k above
I found, I believe, the equations for the laminae:
For the triangle,
[itex]y=4-\frac{4}{3}x[/itex]
For the semicircle,
[itex]x^2+y^2=1[/itex]
And the horseshoe,
[itex]x^2+y^2=25[/itex]
[itex]x^2+y^2=9[/itex]
Welp, I'm lost. As I said, I can figure it out easily using the calculus 2 method of [itex]x_{cm}=\frac{M_y}{M}[/itex], but I'm not sure the way he wants it done.
What's the density formula? Is it just the equations I listed above? What's this (x-k) nonsense?
We've just gone over center of mass with double integrals, so it's a bit peculiar to see this project feature only one integral. I went over that in calc 2 – and as a result, know how to calculate it that way.
However (as you will see in the formula below), I don't think he wants me to calculate it that way. I have no idea how to proceed and the book is of absolutely no help.
Anything you could do to help me out would be fantastic.
And here's the problem:
Homework Statement
There are three type of laminae we are trying to find:
1) A triangle with sides 3", 4", and 5" respectively.
2) A semicircle
3) A "horseshoe", i.e. a half-ring, whose inner and outer edges are composed of circular arcs of radii 3" and 5" respectively.
Definition 1:
The first moment of a planar mass distribution about a line l is the integral over the region of the (area) density σ(P) times the distance of P from the line. For example, the first moment about the line x=k is
[itex]M_k=∫_R\left(x-k\right) σ\left(P\right) dA[/itex]
Definition 2:
In order for a line to be a balancing line, the first moment about this line must be zero.
Homework Equations
See the M_k above
I found, I believe, the equations for the laminae:
For the triangle,
[itex]y=4-\frac{4}{3}x[/itex]
For the semicircle,
[itex]x^2+y^2=1[/itex]
And the horseshoe,
[itex]x^2+y^2=25[/itex]
[itex]x^2+y^2=9[/itex]
The Attempt at a Solution
Welp, I'm lost. As I said, I can figure it out easily using the calculus 2 method of [itex]x_{cm}=\frac{M_y}{M}[/itex], but I'm not sure the way he wants it done.
What's the density formula? Is it just the equations I listed above? What's this (x-k) nonsense?