Tallest free standing structure -- update

The problem with shared blogs (especially with your better half that also happens to be a better writer) is that you can't be lazy about it. My hurried posts the other day invited hands-on-hip-take-it-seriously looks. So here I am atoning for my sins. Grumbling aside, here's my take on the tallest mountain part.

Assuming the mountain isn't a single giant rigid body, its being held in place by friction between the rocks. That puts a limit on the maximum angle that the side of the mountain can have. The co-efficient of friction, u, is related to the angle of the slope (call it s) by tan(s) = u. That means the ratio of the height of the mountain to half its base is also u.

As the mountain gets taller, its base has to become wider and at some point, the curvature of the earth will kick in, thereby limiting the maximum width of the base. Beyond that the mountain "falls off the sides of the earth".

In the picture above, we see that the angle formed at the center of the earth is the same as the angle of the slope (s). r is the radius of earth, h the height of the mountain. Now cos(u) is r/(r+h). Also, u = tan(u) = sin(u)/cos(u).

We know (1 + tan^2(u)) = 1/(cos^2(u). That gives us (1 + u^2) = (1 + h/r)^2. The average r for earth is 6378.1 km. Lets say u is 0.28, that gives max h ~= 245km. Thats quite large and twice over what is considered the edge of space (100km).

Above is of course a rough calculation and there might be aspects I have ignored that limit the maximum height to a lower value (let me know if you know of any)!