Let us consider the following diagram.
From questions,
Torque on trunk of tree = \(\frac{Y\,πr^4}{4\,R}\)
[r = radius of tree, R = radius of curvature of bent surface]
When the tree is about the buckle,
Wd = \(\frac{Y\,πr^4}{4\,R}\)
If R > > h, then the centre of gravity is at a height l = \(\frac{1}{2}\)h from the ground.
From ∆ABC
R2 = (R – d)2 + (\(\frac{1}{2}\)h)2
If d < < R
R2 = R2 – 2Rd + \(\frac{1}{4}\)h2
∴ d = \(\frac{h^2}{8R}\)
If ω0 is the weight/volume
\(\frac{Y\,πr^4}{4\,R}\) = W0(πr2h)\(\frac{h^2}{8R}\) [∵ Torque is due to weight]
or h = (\(\frac{2Y}{W_0}\))\(\frac{1}{3}\) r\(\frac{2}{3}\)