Question

2. (10) Model your tibia as a long cylindrical bone; measure its approximate length and estimate its diameter. When you stand on one leg, how much total shortening occurs in the length of your tibia? The answer to this problem is very person-specific. You will have to use your engineering judgment (and a ruler) to make this calculation. (See Appendix B for the Young's modulus of bone; pay close attention to units.) [ans: amount of shortening will vary with each personH

Answer 1

We have following information

Length of (cylndrical) tibia; l = 36 cm = 0.36 m

Dia of (cylndrical) tibia; d = 1.4 cm = 0.014 m

Radius of tibia ; r = d/2 = 0.007 m

We need following more information to calculate Shortening in Tibia (Strain). You may use the final derived formula to get the value in number.

a. m= Mass of your body (You can use you mass in kg)

b. Y = Young's modulus of bone (As per question, please use Appendix B to refer it) in N/m²

If Y is in different unit, you may use below conversion rates:

1 pascal (1Pa) = 1 N/m²= 1.450377×10⁻⁴ psi ;

1 Giga Pa = 10⁹ N/m²

Now, based on above information we can have following values

Force on tibia; F = m*g; where m is mass of your body

Stress on Tibia;σ  = F/ (cross section area) = F/ (π * r²) = F / (π * d² /4)

If we denote Strain with "x"

Young's modulus of bone, Y = Stress/Strain =σ/x = F / (π * d² /4) / x

Strain; x = F / (π * d² /4) /Y = m*g / (π * d² /4) /Y

Now, Strain is calculated by Ratio of Change in length & Original length

SO, Change in length = Strain * Length of tibia

= x * l

= m*g / (π * d² /4) /Y *l (unit in meter)

(Please use values in SI unit, m = mass in kg, g = 9.8m/s², d = 0.014m, l = 0.36m, Y in N/m² )