Finding Centroid of 2D Shapes & Parabolas

In summary, the conversation discusses the concept of finding the centroid of a 2D shape, which is the point at which the shape balances. It is found by dividing the shape into two parts with equal areas or moments. The conversation also touches on using calculus to find the centroid of a parabola and mentions the concept of center of mass.
  • #1
BlackWyvern
105
0
I think I'm right when I say that the centroid of a 2D shape is found by the intersection of the lines that separate that shape into two shapes of equal areas.
Is that correct? I don't want to (for now) think about it in terms of moments and integrals, because frankly, it's a little confusing.
It would make sense if that's the case.

Then lastly, say we have a parabola, just a standard y = x^2. It's centroid will be found on the y axis, but the exact value is only able to be determined by calculus. I think I'm correct when I say that a section of parabola (made with a horizontal cut) will be similar to the parabola before the cut. Also as a result of the similarity, a parabola will take up the same amount of space for a given rectangle with vertexes on (0, 0) and (x, y).

Using this definition, we can say that the centroid of a parabola that extends to x = 10, y = 100 is found by this method:

[tex]x = 10[/tex]
[tex]y = x^2 = 100[/tex]
[tex]A_{rectangle}= xy = x^3 = 1000[/tex]

[tex]A_{underparabola} = \int_{0}^{10} x^2 dx = 333.3333...[/tex]

[tex]A_{rectangle} - A_{underparabola} = A_{parabola}[/tex]
[tex]1000 - 333.333... = 666.666...[/tex]

[tex]A_{parabola} / A_{rectangle} = P:A = 0.666...[/tex]

P:A is the ratio this parabola takes of it's envelope rectangle (should be constant for all values of [tex]\infty > x > 0 [/tex]
Now the area of the parabola is halved (which gives the area of the lower, parabola shaped section:

[tex]666.666... / 2 = 333.333...[/tex]
[tex]333.333... / P:A = A_{smallrectangle} = 500[/tex]

[tex]500 = xy = x^3[/tex]
[tex]x = \sqrt[3]{500}[/tex]
[tex]y = x^2[/tex]
[tex]y = 500^{2/3}[/tex]

y ~ 63
Centroid = (0, 63)

I'm pretty sure this is correct, but can someone who's a bit more senior confirm for me?
Thanks.
 
Physics news on Phys.org
  • #2
BlackWyvern said:
I think I'm right when I say that the centroid of a 2D shape is found by the intersection of the lines that separate that shape into two shapes of equal areas.
Is that correct?
No. The lines must divide the shape into parts of equal moment about the line.
I don't want to (for now) think about it in terms of moments and integrals, because frankly, it's a little confusing.
Oh well...

Think of it as finding the center of mass of the object.
 
  • #3
For the parabola example then, how would you do it?
 
  • #4
BlackWyvern said:
For the parabola example then, how would you do it?
See: http://mathworld.wolfram.com/ParabolicSegment.html" [Broken]
 
Last edited by a moderator:

1. What is the centroid of a 2D shape?

The centroid of a 2D shape is the geometric center or average position of all the points in the shape. It is often referred to as the "center of mass" or "center of gravity" of the shape.

2. How is the centroid of a 2D shape calculated?

The centroid of a 2D shape can be calculated by taking the average of the x-coordinates and the average of the y-coordinates of all the points in the shape. This can be expressed mathematically as (x̅, y̅), where x̅ is the average of all the x-coordinates and y̅ is the average of all the y-coordinates.

3. What is the significance of finding the centroid of a 2D shape?

Finding the centroid of a 2D shape can provide valuable information about the shape's properties, such as its balance and stability. It is also used in various applications, such as engineering and architecture, to determine the center of mass for structural analysis.

4. Can the centroid of a 2D shape fall outside of the shape?

Yes, it is possible for the centroid of a 2D shape to fall outside of the shape. This typically occurs when the shape is irregular or has varying densities. In these cases, the centroid may not be located at a point within the shape itself, but it will still represent the average position of all the points in the shape.

5. How is the centroid of a parabola calculated?

The centroid of a parabola is calculated by using the formula (1/2, 2/3) * c, where c is the distance between the vertex and the focus of the parabola. This can be derived from the properties of a parabola, such as its line of symmetry and the location of its focus.

Similar threads

  • Engineering and Comp Sci Homework Help
Replies
1
Views
762
  • Introductory Physics Homework Help
Replies
3
Views
858
  • Introductory Physics Homework Help
Replies
3
Views
735
  • Calculus and Beyond Homework Help
Replies
9
Views
698
  • Engineering and Comp Sci Homework Help
Replies
2
Views
2K
  • Calculus and Beyond Homework Help
Replies
1
Views
416
Replies
4
Views
1K
  • Calculus and Beyond Homework Help
Replies
6
Views
852
  • Linear and Abstract Algebra
Replies
5
Views
985
  • Calculus
Replies
4
Views
2K
Back
Top