BME Application: Bones naturally have less dense material towards the center with more dense material towards the periphery. In this problem, we will mathematically analyze how this structural characteristic aids us in our day$-$to$-$day movement.

Assume we have two bone samples$-$one can be modeled as a solid cylinder with a radius of $2\mathrm{.}8$ cm $,$ and the other one can be modeled as a hollow cylinder with an outer radius of $2\mathrm{.}8$ cm and an inner radius of $2\mathrm{.}2$ cm $.$ Both samples experience an internal bending moment of $\backslash (\backslash$mathrm$\{$M$\}=350\backslash$mathrm$\{$Nm$\}\backslash ).$

a$)$ Calculate the maximum bending stress in the bone modeled as a solid cylinder.

b$)$ Calculate the maximum bending stress in the bone modeled as a hollow cylinder.

c$)$ Calculate the area of bone for the bone modeled as a solid cylinder.

d$)$ Calculate the area of bone for the bone modeled as a hollow cylinder.

e$)$ For both bones, find the ratio of maximum bending stress to the area. Which one is greater? $($Note that the bone with the bigger ratio of maximum bending stress to area is a more 'efficient' use of bone material.$)$

N$.$B$.$:

$-$ The formula for calculating inertia of a sold cylinder about its cylinder axis is $\backslash ($ I$=\backslash$pi $/4*$ R$^\{4\}\backslash ).$

$-$ R is the radius of the solid cylinder.

$-$ The formula for calculating inertia of a hollow cylinder about its cylinder axis is $\backslash (\backslash$mathrm$\{$I$\}=\backslash$pi $/4*\backslash$left$(\backslash$mathrm$\{$Ro$\}^\{4\}-\backslash$mathrm$\{$Ri$\}^\{4\}\backslash$right$)\backslash )$

$-\backslash ($ R$\_\{$i$\}\backslash )$ is the inner radius of the hollow cylinder and $\backslash ($ R$\_\{$o$\}\backslash )$ is the outer radius of the hollow cylinder.