(1)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)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 compliancedetermines 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)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 itsenergy--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?
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 isallometric 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 https://docs.google.com/spreadsheets/d/19nlj0Ja0_CajV-8JA4qWRp4dZdBT-M4eqKNirmkHrHk/edit?usp=sharing. This document is pre-populated with some real data I collected, but to continuallyimprove 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. Select the two columns of numbers you wish to plot (Height and Foot Area).
2. Click on the Charts tab & select Scatter plot
3. Select the resulting plot and click on the Chart Layout tab &Trendlinesubtab.
4. 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.
