Physics 100

Scaling in Bones

Galileo made an interesting suggestion about how bones should scale.  An animal twice as big would weigh 8 times as much, because mass goes as , where L is the linear dimension.  Now suppose the bigger animal were simply a scale model of the smaller one.  Then its bones would have a cross-sectional area only 4 times greater, because area goes as .  This would make the larger animal relatively weaker than the small one, because in Galileo’s mind the strength of the bone is proportional to its cross-sectional area.  Reasoning that animals are in many ways scale models of each other, and yet Nature would not permit larger animals to be too weak, Galileo suggested that the cross-section of the bones should be 8 times greater in an animal twice as big, not 4 times, so that the bones’ diameter would be adequate to the animal’s weight.  This implies that when the bones’ length goes up by a factor 2, the diameter goes up by a factor , and in general, the diameter D should scale like .   We thus have the prediction, for some constant k,

Galileo’s exposition was so persuasive that this idea wasn’t really checked experimentally until quite recently, even though it is very easy to do.  (We will do it ourselves!)

You will have access to a number of mounted skeletons (delicate structures, so please be VERY careful).  With tape measure and calipers measure the length L and diameter D, across species, of some bone (or bones).  You may have to think a bit to identify the homologous bone in some species.  Don’t omit the horse out in the corridor in the basement of Clapp, or the extinct giant sloth on the 4th floor landing, west side of Clapp!

We will analyze these data in two different ways.  Both ways rely on writing the expected relation between L and D as

This way of writing it follows by taking logarithms in the power law relation we began with, further up the page.  Thus if we plot log(D) vs. log(L), we expect a straight line with slope 3/2.  More generally, if it were some other power law, we would still get a straight line, but with a different slope, corresponding to the power.

It is so common to analyze data looking for power laws that there is special graph paper, log-log paper, to make it easier to do.  When you plot values on log-log paper, using the axes at the edge, you are really plotting the logarithms of the values (just what we want to do).  Plot your data on log-log paper, and draw what conclusions you can.

We will be using the generally useful spreadsheet program Excel many times in this course, and this is a chance to try it out.  Good instructions for using Excel, written by Fawn Huisman, are available, both on paper and online.  We will also give a quick and sketchy tutorial here.

Open Excel, and put your data into a neat table with a column for L and a column for D.  It is a good idea to label the columns – just type a name in the box at the top of the column, such as “L (cm)”, or some such thing.  Now we actually want the logarithms.  These will go in new columns (perhaps headed “log(L)”, etc.)   If the data for L occupy cells A2:A11, for example, then in cell C2 you could type “=log(A2)”.  Excel does the computation.  Now – and this is the neat part – you can use the mouse to drag the C2 box down through C11.  Excel will assume you want the corresponding information in each cell, and with luck you will find log(L)  in column C.  Do the corresponding thing to get a column with log(D).  Now to plot the logarithms, use INSERT from the Menu, and CHART.  Choose (XY) SCATTER for the type of chart.  Click NEXT to go on to specify more about the scatter plot.  Find the “SERIES” tab on the next screen, and specify the columns with the logarithms for the x and y data.  To do this select the x window, then highlight one of the columns in the spreadsheet.  Select the y window and highlight the other column.  Click NEXT and add labels and a title to the graph.  Then FINISH.   To fit a line to the finished plot, RIGHT CLICK on any datapoint in the plot, and ask for a TRENDLINE.  Also go to OPTIONS and ask for the equation of the trendline to be shown.  That’s it!  The rest is thinking and interpretation.

Scaling in Bones worksheet:

Which bone(s) did you choose?

Data table:

Sketch of graph, with axes labeled, and the equation of the best fit line:

Write a paragraph saying what conclusions you can draw from this measurement.  Be sure to comment on the slope of the best fit line, and also comment on the uncertainty in this quantity, which you can estimate by drawing other plausible lines.   In light of this result, what can you say about Galileo’s original suggestion?