will use this relationship to determine how a bone's cross$-$sectional shape adapts to changes in

mechanical loading.

Questions:

$2-1.(2$ points$).$ For a bone whose cross$-$section is a circular tube with an outer radius of $18$ mm and an

inner radius of $12$ mm $,$ find the section$\text{'}$s polar moment of inertia $($ J $,$ in units of $m{m}^4).$

$2-2.(2$ points$).$ For the bone geometry in question $3-1,$ compute the shear stress $(,$ in units of MPa$)$ at

the periosteal surface and the shear stress at the endosteal surface for an applied torque of $20,300$

$N*mm.$

$($continued on next page$)$

$2-3.(7$ points$).$ Consider a person whose femoral diaphysis has the geometry provided in question $3-1.$

The cortical bone in their femur is transversely isotropic and has an elastic modulus $E=17$GPa and a

shear modulus $G=3\mathrm{.}3$GPa. The person's daily activities result in $10,000$ torsional cycles per day with

a torque magnitude of $20,300N*mm.$ The energy stress in a transversely isotropic material under a

shear stress of magnitude $$ is

$\frac{?}{\text{b}\text{a}\text{r}}()=(\frac{E}{G}{)}^{\frac{1}{2}}$

Based on the modeling rate relationship provided above, find the modeling rates $(r^{},$ units of

$\frac{m}{?}$ day $)$ on the periosteal and endosteal surfaces of the femoral cross$-$section. At the endosteal

surface, is bone being resorbed or formed? At the periosteal surface, is bone being resorbed or

formed?

$2-4.(7$ points$).$ The same person starts a vigorous exercise program and performs an additional $3000$

loading cycles per day at a torque of $30,000N*mm($in addition to the $10,000$ cycles at $20,300$

$N*mm).$ What are the surface modeling rates $(r^{},$ units of $\frac{m}{?}$ day$)$ on the periosteal and endosteal

surfaces now? Will the outer diameter of the bone increase or decrease? Will the endosteal

diameter increase or decrease?

$2-5.(7$ points$).$ Instead of starting an exercise program, the person cuts out carbs and loses $20\%$ of his

body weight. The femur now experiences $10,000$ cycles$/$day of a torque that is $80\%$ of the previous

$20,300N*mm$ torque. What are the surface modeling rates $(r^{},$ units of $\frac{m}{?}$ day$)$ on the periosteal

and endosteal surfaces now? Will the outer diameter of the bone increase or decrease? Will the

endosteal diameter increase or decrease?