Physics 103

**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 4^{th} 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?

OPTIONAL: if you wish, please say what you liked or didn’t like about this lab, and how it might be improved.