Is your Physics Textbook Lying? Linear & Nonlinear Springs *
Difficulty  
Time Required  Very Short (≤ 1 day) 
Prerequisites  An introductory high school physics course would be helpful (but not required) for this project. 
Material Availability  You may be able to find some springs at home, for example by disassembling pens or toys. If you cannot find any at home, many online retailers such as Amazon.com and specialty companies will have springs. 
Cost  Very Low (under $20) 
Safety  No issues. 
Abstract
Spoiler alert: Your physics textbook might contain an inaccurate equation. Are you shocked? Let us explain — many questions in your physics textbooks are simplifications of how things behave in the real world. For example, in physics textbooks, springs are usually modeled with the equation Force = stiffness x displacement:Equation 1:

This equation describes a linear spring — if you plot a curve showing force vs. displacement for such a spring, it will be a straight line with a slope of k. That straight line denotes a "linear relationship" between force and displacement, one in which they are directly proportional to each other.
However, not all springs behave this way. Equation 1 is an approximation. It may be a very good one for certain springs made from certain materials, or springs that don't stretch very much. More generally, a nonlinear spring can have a force vs. displacement curve that is not a straight line (indicating a nonlinear relationship between force and displacement). Because the slope of that curve is not constant, it does not make sense to talk about a "spring constant." Instead we refer to the slope as the stiffness. If you have taken calculus, that means the stiffness can be expressed as the derivative of the force vs. displacement curve:
Equation 2:
If you have not taken calculus, do not worry about Equation 2 — just remember that the stiffness of a nonlinear spring is the slope of the force vs. displacement curve at any given point. For a more indepth discussion of linear and nonlinear springs, see the Science Buddies' Linear & Nonlinear Springs Tutorial.
In this experiment, you will test a variety of springs (what kind and how many is up to you) to see if they behave in a linear way (where their force and displacement are directly proportional). You will need to conduct experiments to gather data and create force vs. displacement curves for each spring (hint: one good way to do this is to hang weights from the springs while holding them next to a ruler, so you can measure displacement). If the resulting force vs. displacement is linear, then Equation 1 is a good approximation for your spring. If not, then the spring is nonlinear, and you can use Equation 2 to calculate its stiffness as a function of displacement.
Keep in mind that many things can behave like springs even if they do not look like springs. Anything that returns to a neutral position after it is stretched can be treated like a spring. For example, rubber bands exert a force when they are stretched, and return to their original size when they are released, so they behave like springs. However, this does not guarantee that they are linear springs. How can you tell if they are? What experimental procedure can you devise to apply to your collection of springs to determine if they are linear or nonlinear?
Credits
Ben Finio, Ph.D., Science Buddies
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