CM of triangle with integration

In summary, the conversation discusses how to calculate the center of mass of an isosceles triangle using integration, with a constant density. The formula Rcm = integral(xdm) / integral(dm) is mentioned and the attempt at a solution includes defining the boundaries of integration. The conversation also mentions that the triangle is made of two right triangles and that the integration limit for y is simply b.
  • #1
drugcpp
2
0

Homework Statement



I need to calculate center of mass of this isosceles triangle using integration:
42pw9.png

the density is constant.

Homework Equations


Rcm = integral(xdm) / integral(dm)

The Attempt at a Solution


I know how to begin:
dm = density * dx * dy;
x = double-integral(x*dx*dy) / integral(dy*dx);
y = double-integral(y*dx*dy) / integral(dy*dx);

I have problem with defining the boundaries of integration.
let's say that c is the base and b is the height of this triangle.
I think that -c/2 to c/2 are the boundaries for x, but I have no idea what the boundaries for y are. I know how to solve similar problem with right triangle, a boundary for y there is defined by function b-(b/a)x. Any help?
 
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  • #2
The triangle is made of two right triangles.

ehild
 
  • #3
Really?! OMG!
I see that myself. What I asked is integration limit. I've solved it myself already, the limit is simply b.
 

FAQ: CM of triangle with integration

What is the centroid of a triangle?

The centroid of a triangle is the point of intersection of its three medians. A median of a triangle is a line segment that connects a vertex to the midpoint of the opposite side. The centroid is often referred to as the "center of mass" or "center of gravity" of the triangle.

How is the centroid of a triangle calculated?

The centroid of a triangle can be calculated by finding the average of the x-coordinates and the average of the y-coordinates of the three vertices. This point is known as the centroid point and represents the coordinates of the centroid.

What is the significance of the centroid of a triangle?

The centroid of a triangle has several important properties. It is the center of mass of the triangle, meaning that if the triangle is cut out of a sheet of uniform material, the centroid is the point where the triangle can be balanced on a needle. It also divides each median into a 2:1 ratio.

How is integration used to find the centroid of a triangle?

Integration is used to find the centroid of a triangle by calculating the area of the triangle and the coordinates of its centroid. The area can be found using the formula A = (1/2)bh, where b is the base and h is the height. The coordinates of the centroid can then be found by taking the integral of the triangle's x and y coordinates over the area of the triangle.

Can the centroid of a triangle be outside the triangle?

Yes, it is possible for the centroid of a triangle to be outside the triangle. This can occur when the triangle is not a regular shape or when it is an obtuse triangle. In these cases, the centroid will still follow the same rules and properties, but it may not be located within the triangle itself.

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