What is ceter of gravity concept and how to calculate it for a triangle
This is a pure math question - it should be moved.
To answer the question you need to know elementary calculus.
The centre of gravity these days is referred to as the Centre of Mass of an object.
To solve physics problems regarding motion, equilibrium etc it is necessary to consider the mass of objects.
However complicated the shape of a real object may be its mass can be considered as concentrated at a single point.... the centre of mass. The force of gravity on the object acts through the point known as the centre of mass.
One consequence of this is that an object can be balanced by a support placed at, or passing through, the centre of mass.
There are mathematical techniques for calculating the position of the centre of mass (applied physics) and there are experimental techniques for locating the centre of mass.
Work it out for a simple 45 degree right triangle by integrating. Then relate it to the vertices. The result will generalize though you won't have a proof, just a result.
There is no need to calculate integrals. It is easy to see that center of mass of a triangle lies on its median. As a conclusion - the medians must intersect in one point, and this point is the center of mass.
Yeah, I don't see why folks go on about needing calculus when this works just fine.
Let me point out that what people are discussing here is more properly called the "centroid", a purely geometric concept. the "center or mass" and "center of gravity", which are physics concepts, require a density function. If the density is a constant, then the "center of mass" and "center of gravity" are the same as the "centoid". Yes, the centroid of a triangle is just the point whose coordinates are the average of the corresponding coordinates of the vertices. For a general polygon, it is a little more complicated- you can divide the polygon into triangles, find the centroid of each triangle, the find the weighted average of those, the "weighting" being by the area of the triangles. For more general figures, with curved sides, you will need to integrate.
Same thing in three dimensions. The coordinates of the centroid of a tetrahedron are the average of the corresponding coordinates of the four vertices. Any polyhedron can be divided into tetrahedrons and the centroid is the weighted average of the centroids of the tetrahedron (weighted by their volume) while centroids of figures with curved surfaces require integration.
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