# Spring Constants of Body Parts

When tissues are put under within their they exert a that is proportional to the . They also return back to their original size and shape when the force causing the deformation is removed. This exactly how springs behave, so we can model tissues, and any other material within its , as a collection of springs.

Consider modeling the humerus bone as a spring, as depicted in the image above. We can think of compressing a bone that has twice the of the humerus as equivalent to compressing two of the original springs at the same time, which would require twice the applied force to create the compression distance (). Therefore that bone would have twice the of the humerus. We can also think of a bone that is twice as long as the humerus as equivalent to compressing two of the original springs placed end-to-end. Each spring will only have to compress half of the total distance, so that would require only half the force to create the same total compression distance. Now we can see that the size of an object affects the . As a result, the force required to achieve a particular compression or stretch is different for different sized objects, even when they are made of the same material. In order to study the elastic properties of a material such as bone, independent of how big the bone is, then we need to remove the effect of size. That is why we have been working with , , and rather than , , and .

# Relating Spring Constant and Elastic Modulus

We have learned that the tells us how much force is required to stretch a spring a certain distance. We have also learned that the tells us how much is required to cause a certain . It seems like these concepts are very similar, but not quite identical, which is true. The stress and strain relation within the is really just after accounting for the amount and shape of the material being deformed, which allows us to analyze the material itself independent of the size of the object. Let’s see how elastic modulus and spring constant are related.

If we start with the elastic stress-strain relation:

(1)

And then replace stress and strain with their definitions from previous chapters:

(2)

We can rearrange to get force and stretch distance on opposite sides

(3)

Which looks exactly like Hooke’s Law if the spring constant of an object is just all the stuff sitting in front of the stretch distance:

(4)

Now we can see that the elastic property of materials causes them to behave like springs. The human body has adapted to take advantage of the springy nature of tissues to walk more efficiently, jump higher, and generally improve performance in many activities. Upcoming units will help us to understand the physics behind these adaptations.

### Everyday Example

Earlier we stated that a typical would stretch by about the width of a human hair when an additional 480 **lbs** (2135 **N**) of tension was applied after . How did we arrive at that surprising figure?

First we looked up the elastic modulus of the Achilles tendon and found the value to be 1.2 **GPa**.^{[1]}

Then we looked up the typical length and diameter of the Achilles tendon and found 0.15 **m **for the length and 0.018 **m** for the minimum diameter. ^{[2]}

We approximated the by assuming the tendon cylindrical with circular cross-section:

Then we inserted the , original length, and area into the equation for spring constant:

(5)

Now that we have the spring constant for a typical tendon, we used to relate the tendon force and the stretch distance.

Dividing both sides by isolates the stretch distance, which is what we want. Then we insert the k and F values, remembering to use for our force unit to match the units on the we calculated.

We can write our answer above in as or using a . Typical human hair is also a few hundred as you can see from this 2000x magnification image of a human hair produced by a scanning electron microscope (SEM).

- "In vivo human tendon mechanical properties" by Constantinos N Maganaris and John P Paul, U.S. National Library of Medicine, National Institutes of Health ↵
- "Achilles tendon: functional anatomy and novel emerging models of imaging classification" by Angelo Del Buono, Otto Chan, and Nicola Maffulli, U.S. National Library of Medicine, National Institutes of Health ↵
- "Human Hair 2000X" by MUSE via Wikimedia Commons [CC BY-SA 3.0 (https://creativecommons.org/licenses/by-sa/3.0)] ↵

a physical quantity that expresses the internal forces that neighboring particles of material exert on each other

the range of values for stress and strain values over which a material returns to its original shape after deformation

a force that tends to move a system back toward the equilibrium position

change in position, typically in reference to a change away from an equilibrium position or a change occurring over a specified time interval

The cross-sectional area is the area of a two-dimensional shape that is obtained when a three-dimensional object - such as a cylinder - is sliced perpendicular to some specified axis at a point. For example, the cross-section of a cylinder - when sliced parallel to its base - is a circle

measure of the stiffness of a spring, defined as the slope of the force vs. displacement curve for a spring

the measure of the relative deformation of the material

measures of resistance to being deformed elastically under applied stress, defined as the slope of the stress vs. strain curve in the elastic region

any interaction that causes objects with mass to change speed and/or direction of motion, except when balanced by other forces. We experience forces as pushes and pulls.

region of the stress vs. strain curve for which stress is proportional to strain and the material follows Hooke's Law

the restoring force exerted by a spring is equal to the displacement multiplied by spring constant

a tough band of fibrous tissue that connects the calf muscles to the heel bone

non-permanent re-alignment of substructures (fibers) in a material that results in non-linear behavior at stress values less than the yield stress.

a way of writing very large or very small numbers. A number is written in scientific notation when a number between 1 and 10 is multiplied by a power of 10.

a unit prefix that precedes a basic unit of measure to indicate a multiple or fraction of the unit