General Centre of mass derivation

[shaggySS]

Hello,
It's my first question here. So I'll try to give as much as information as I know

Actually I am stuck with a problem of centre of mass derivation of a triangle with its base on the X axis and symmetric about it.
The base is b and height is H

As far as I know I have to imagine it as a couple of 1d bodies and integrate them.

Thanks
SAGNIK

 

[shaggySS]

Since its symmetric about the X axis the y coordinate needs to be found only.

 

[HallsofIvy]

The y- coordinate of the center of gravity of an object with density function $\delta(x,y)$ is
$$\frac{\int y \delta(x,y) dxdy}{\int \delta(x, y) dxdy}$$
In particular, if $$\delta$$ is a constant, it can be factored out of the two integrals and cancelled- that is the "center of gravity" is just the
geometrical "centroid". Here, we can set up a coordinate system so that the origin is at the center of the base of the isosceles triangle, base
along the x- axis, height along the y-axis. The equations of the two slant sides can easily be determined.