Question

A man grows into a giant such that his linear dimensions increases by a factor of $9$. If his density remains same, then stress on the leg will increase by a factor of
(Assume cubical shape of man with $L$ as edge length).

Answer 1

Let the initial mass of man is $m$ and cross-sectional area of his legs is $A$ then mass is given as,
$m=\rho V=\rho L^3$ and $A=L^2$
$[$ assuming cubical shape of man having edge length $L]$
External load acting on man is his weight $\left(mg\right)$
$\Rightarrow \text{stress}=\frac{\text{force}}{\text{area}}=\frac{mg}{A}=\frac{\rho L^{3}g}{L^2}$
$\Rightarrow {\sigma }_i=\rho Lg....\left(1\right)$
When man's linear dimensions increases by a factor of $9$, Let the final mass of man is $m^{\prime }$ and area of leg is $A^{{}^{\prime }}$
$\Rightarrow m^{{}^{\prime }}=\rho V^{{}^{\prime }}=\rho (9L{)}^3=9^3\rho L^3$ and $A^{{}^{\prime }}=(9L{)}^2=9^2{L}^2$
$\Rightarrow \text{stress}=\frac{\text{force}}{\text{area}}=\frac{m^{{}^{\prime }}g}{A^{{}^{\prime }}}=\frac{9^3\rho L^{3}g}{9^2{L}^2}$
$\Rightarrow {\sigma }^{\prime }=9\rho Lg....\left(2\right)$
From Eq.$\left(1\right)$ and $\left(2\right)$,
$\Rightarrow \frac{{\sigma }^{\prime }}{\sigma }=9$
$\therefore {\sigma }^{\prime }=9\sigma$
Hence stress in the leg will increase by a factor of $9$.