A broadleaf tree species

PHYSICS · Multi-part · Fermi University Physics · Fluids & Continuum Physics · Biophysics & Medical Physics · Bioelectricity, the senses and the physics of the body

Forces, Machines and Signals, Ch Bones, blood and body size

Problem

A broadleaf tree species, whose trunk thickens as it grows, is modelled as a uniform circular cylindrical trunk of diameter

d

and length

L

, of density

\rho

and Young's modulus

E

, both the same at every age. The trunk is treated as a slender column loaded only by its own weight, buckling elastically (Euler buckling) when the self-weight reaches the critical load, with the same safety margin against buckling at every size (elastic similarity). A column loaded by its own weight buckles at a load of the same form as the end-loaded one, with a prefactor of about

7.8

in place of

\pi^2

, so only proportionality matters here. ((a)) Starting from the Euler critical buckling load

F_c = \pi^2EI/L^2

with

I = \pi d^4/64

, and the self-weight

W = \rho g(\pi d^2/4)L

, show that requiring

F_c

to remain proportional to

W

as size changes gives

d \propto L^{3/2}

. ((b)) Using

M \propto d^2L

, show that this implies

L \propto M^{1/4}

and

d \propto M^{3/8}

. ((c)) A sapling of mass

5.0\,\mathrm{kg}

grows into a mature tree of mass

1280\,\mathrm{kg}

(a mass increase of a factor of

256

). Using the scaling of part (b), find the ratio of mature to sapling trunk length and the ratio of mature to sapling trunk diameter.

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