Find Centroid of Region: No Integration Needed

In summary, the conversation discusses finding the centroid of a region shown, without using integration. The suggested method is to find the centroids of the rectangles and triangles within the region and then use the formula for additivity of moments. The formula for the centroid of a triangle is given and the importance of organizing calculations is emphasized. The conversation concludes with the individual centroids being confirmed as correct and the final answer for the overall centroid being found.
  • #1
Jbreezy
582
0

Homework Statement



Wondering if I did this right. Find the centroid of the region shown , not by integration, but by locating the centroids of the rectangles and triangles and using additivity of moments.

Homework Equations



I will give you the coordinates since I can't draw it.
For the rectangle (-1,0), (0,0) ,(-1,2), (0,2) and for the triangle I have (0,0), (2,0) and (0,2)

The Attempt at a Solution



So I did x Bar = M(triangle) + M(square) / (Area triangle + area square)
So to get M(triangle) I did xbar(area triangle) = (1)(4/2) = 2

I did this for the square too. I got (-1) When I plugged it into my above formula I got x bar = (1/3)

So, I did the same thing but with y. So I got y bar = 4/3

Does anyone agree? I pretty much did it like an example in class.

Thanks!
 
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  • #2
Your triangle has a base of 2 and a height of 2. According to you, the centroid of this triangle is 1 unit from the origin. Does this look right? What is the centroid of a triangle? There is a formula which you should be using.

Trying to follow your work above is confusing. You should organize your calculations better.

Remember, for a plane figure, the centroid requires 2 numbers to locate it.
 
  • #3
I don't know what it is for a triangle. I think it like 1/3 something. I don't know.
 
  • #4
Well, you could look it up, if it's not too much trouble. But what you guessed definitely ain't it.
 
  • #5
The centroid of a "simplex" in n dimensions (triangle in 2 dimensions, tetrahedron in 3 dimensions, a polyhedron with n+1 vertices in n dimensions) has coordinates the average of the coordinates of the vertices. If a triangle in the plane has vertices [itex](x_1, y_1)[/itex], [itex](x_2, y_2)[/itex], and [itex](x_3, y_3)[/itex], then its centroid is at
[tex]\left(\dfrac{x_1+ x_2+ x_3}{3}, \dfrac{y_1+ y_2+ y_3}{3}\right)[/tex]

The centroid of more complex figures can be found by "triangulation"- dividing the figure into triangles, finding the centroid of each triangle, then taking the "weighted" average, weighted by the areas of the triangles. That is what your text means by "additivity of moments".
 
  • #6
Using your formula the centroid the triangle was (2/3,2/3)
and the centroid I got for the square was (-1/2), 1
This seems reasonable to me. Then I summed the x components and the y components and divided each by the total area of both objects which is 4. So for my final answer I ened up with (5/12, 1/24) Seems pretts low.
 
  • #7
So I know my two centroid of the individual objects are OK. But I'm confused on the next part.
 
  • #8
Solved it. Thx
 

1. What is the centroid of a region?

The centroid of a region is the point that represents the center of mass of that region. It is the point at which the region would balance if it were placed on a pin.

2. How is the centroid of a region calculated without integration?

The centroid of a region can be calculated without integration by using the geometric properties of the region, such as symmetry and parallel axis theorem. This method is known as the method of moments.

3. What are the steps involved in finding the centroid of a region without integration?

The steps involved in finding the centroid of a region without integration are as follows:

  • Identify the geometric properties of the region, such as symmetry and parallel axis theorem.
  • Choose a coordinate system and label the axes.
  • Divide the region into simpler shapes, such as rectangles or triangles.
  • Find the centroid of each smaller shape using their known formulas.
  • Calculate the overall centroid by using the weighted average of the centroids of the smaller shapes.

4. What are the advantages of using the method of moments to find the centroid of a region?

The method of moments is advantageous because it does not require advanced mathematical knowledge or complex calculations. It also provides a more intuitive understanding of the centroid and its location within the region.

5. In what fields is finding the centroid of a region without integration commonly used?

Finding the centroid of a region without integration is commonly used in fields such as physics, engineering, and architecture. It is also used in computer graphics and image processing to determine the center of an object.

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