Question

Bone has a Young's modulus of about1.8 x 10^10 Pa. Under compression, it canwithstand a stress of about 1.69 x 10^8 Pa be-fore breakingAssume that a femur (thigh bone) is 0.49 mlong, and calculate the amount of compressionthis bone can withstand before breaking.Answer in units of mm.

Answer 1

4.6 mm

Step-by-step explanation:

The Young's Modulus is defined as

`Y=P(L_0)/(\Delta L)`

Where

Y = Young's Modulus

P = pressure (stress) applied

L0 = inital length

ΔL = change in length

Now in our case, we know that the femur is 0.49 m long (L0 = 0.49) and it can withstand max stress of 1.69 x 10^8 Pa ( P = 1.69 x 10^8 Pa), and Young's Modulus for the bone is 1.8 x 10^10 Pa ( Y = 1.8 x 10^10 Pa ); therefore, the above formula gives

`1.8*10^(10)=1.69*10^8*(0.49)/(\Delta L)`

We have to solve the above equation for ΔL, the amount of compression.

Multiplying both sides by ΔL gives

`\Delta L*1.8*10^(10)=1.69*10^8*0.49`

dividing both sides by 1.8 x 10^10 gives

`\Delta L=(1.69*10^8*0.49)/(1.8*10^(10))`

which we evaluate to get

`\Delta L=(1.69*0.49)/(1.8)*(10^8)/(10^(10))`
`\begin{gathered} \Delta L=4.6*10^(-3)m \\ \boxed{\Delta L=4.6\; mm\text{.}} \end{gathered}`

Hence, the amount of compression a femur can withstand is 4.6 x 10^-3 m or 4.6 mm.