You may work on this homework on your own or discuss it with others. If you work through

questions with others, be sure to turn in your own work, at the level at which you understand

it. Do not simply copy answers from or for someone else. Questions such as these will show

up on exams, so be sure to work to understand how to arrive at the answers.

(1) (4 pts) Figure 1 depicts samples from four cylindrical tendons and all are composed primarily of

collagen. Each has been clamped at both ends and a machine is used to gradually pull the ends in

opposite directions (arrows) until the material yields (i.e., experiences internal damage).

a. Which tendon sample(s) will probably require the most tensile force to cause internal damage?

Why?

b. Which tendon sample(s) will probably require the highest stress to cause internal damage? Why?

c. Which tendon sample(s) will probably change length the most before experiencing internal

damage? Why?

d. Which tendon sample(s) will probably have the highest elastic extensibility? Why?

(2) (4 pts) Figure 2 shows the stress-strain curves under tension for four materials in their elastic

regions. Recall that materials have the following properties:

– A material’s elastic modulus (E, “intrinsic stiffness”) determines how much stress (tensile or

compressive) on a material is required to induce a given strain [units of pressure]– A material’s compliance determines how much strain results from an applied stress (compliance

is the inverse of E) [units of 1/pressure]– A material’s elastic strength is how much stress (tensile or compressive) is applied at the point

where the material yields (i.e., stops responding elastically to the applied stress) [units of pressure]– A material’s elastic extensibility is how much strain results at the point where the material yields

(i.e., stops responding elastically to the applied tensile

stress) [units of length]a) List the materials in the order of their elastic

modulus (stiffness), from highest to lowest.

b) List the materials in the order of their compliance,

from highest to lowest.

c) List the materials in the order of their elastic

strength, from highest to lowest.

d) List the materials in the order of their elastic extensibility, from highest to lowest.

(3) (4 pts) The picture below on the left shows an orb web. The red lines are called frame lines and

the blue lines are spiral threads. The frame lines hold the spiral threads in their proper place and

provide a structural framework. The spiral threads intercept flying insects; they slow the insect

down by absorbing its energy–stretching and not breaking.

Which material in the stress-strain plot below (Fig. 3) would be best for a spiral thread? Why?

Which would be best for a frame thread? Why?

Fig. 3

(4) (3 pts) Larger individual humans tend, not surprisingly, to have larger feet. But how exactly

would you expect the surface area of the bottom of an individual’s foot, which must support the

downward force (weight) of the standing person, to scale against body size? Two possibilities are

that scaling would simply be isometric or that scaling is allometric such that stress similarity is

maintained among feet of individuals of different sizes.

a) Assuming full-body isometric scaling among individuals of different sizes, what would you

predict would be the value of the scaling coefficient (b) relating the surface area of the

bottom of the foot to body height?

b) Alternatively, assuming that scaling of feet is allometric such that stress similarity is

maintained among feet of individuals of different sizes (i.e., the stress on the bottom of the

foot is constant among individuals of different height), what would you predict would be the

value of the scaling coefficient (b) relating the surface area of the bottom of the foot to body

height? Assume that the rest of the body scales isometrically.

c) Use real data to see whether the actual relationship between human foot area and human

height appears to be consistent with (a) or (b), above. To do this, go to the google doc

spreadsheet at httpsss://docs.google.com/spreadsheets/d/19nlj0Ja0_CajV-8JA4qWRp4dZdBTM4eqKNirmkHrHk/edit?usp=sharing. This document is pre-populated with some real data I

collected, but to continually improve our data set, please add data from yourself (no need to

add your name, just your data!). Simply enter your sex, height, and the length, maximum

width, and area of the bottom of your right foot (approximated as length X maximum

width). All measurements should be made to the nearest tenth of a centimeter.

You can then use Excel to fit a power function (scaling equation, Y=axb) to these data.

Exactly how to do this will depend on your version of Excel. But the procedure will be

something like this:

1.

2.

3.

4.

Select the two columns of numbers you wish to plot (Height and Foot Area).

Click on the Charts tab & select Scatter plot

Select the resulting plot and click on the Chart Layout tab & Trendline subtab.

Under Trendline Options, select Power & under Options (just to the left) click on

“Display equation on chart” and click OK.

The equation of the power function fitted to your data should be displayed on your graph

and you can now see whether the observed scaling coefficient appears to support the

scaling relationship of (a) or (b) above.

Give the scaling equation you get from Excel and explain your conclusion based on the

scaling coefficient.