Biophysics Problem 34

A sample of bone in the form of a cylinder of cross-sectional area \(1.5 \;cm^2\) is loaded on its upper end by a mass of \(10\; kg.\) Careful measurement with a travelling microscope reveals that the length decreased by \(0.0065\) percent. What is Young's modulus for the specimen?

The most common mistake in the problem occurs in calculating \(\ell/\ell\)

\(\frac{\Delta \ell}{\ell} \times 100 = 0.0065 \\ \frac{\Delta \ell}{\ell}= 6.5 \times 10^{-5}\)

The next most common mistake is in the conversion of area to \(m^2.\)

\(A = 1.5\;cm^2 \times (1\;m/100\;cm)^2 = 1.5 \times 10^{-4}\;m^2\)

Recall Hooke's Law:

\(\frac{F}{A}= Y\frac {\Delta \ell}{\ell}\)

Now solve for \('Y'.\)

If you used \(F = m\; g = (10\; kg) (9.8 \;m/s^2),\) you should have gotten

\(Y = 0.98 \times 10^{10}\) which is approximately
\(Y = 10^{10}\)