Example 5.4 from College Physics, 5.3 Elasticity: Stress and Strain

Calculating Deformation: How Much Does Your Leg Shorten When You Stand on It?

Calculate the change in length of the upper leg bone (the femur) when a 70.0 kg man supports 62.0 kg of his mass on it, assuming the bone to be equivalent to a uniform rod that is 40.0 cm long and 2.00 cm in radius.

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Strategy
The force is equal to the weight supported, or

$F = mg = (62 . 0 kg) (9 . 80 m {/s}^{2}) = 607 . 6 N,$
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and the cross-sectional area is ${πr}^{2} = 1 . 257 \times10^{- 3} m^{2}$ . The equation $Δ L = \frac{1}{Y} \frac{F}{A} L_{0}$ can be used to find the change in length.
Solution
All quantities except $Δ L$ are known. Note that the compression value for Young’s modulus for bone must be used here. Thus,
$\begin{array}{l} Δ L & = & (\frac{1}{9 \times 10^{9} {N/m}^{2}}) (\frac{607 . 6 N}{1. 257 \times 10^{- 3} m^{2}}) (0 . 400 m) \\ = & 2 \times 10^{−5} m. \end{array}$
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Discussion
This small change in length seems reasonable, consistent with our experience that bones are rigid. In fact, even the rather large forces encountered during strenuous physical activity do not compress or bend bones by large amounts. Although bone is rigid compared with fat or muscle, several of the substances listed in Table 5.3 have larger values of Young’s modulus $Y$ . In other words, they are more rigid.

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