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?
Share your story with Science Buddies!
Yes, I Did This Project! Please log in (or create a free account) to let us know how things went.Credits
Ben Finio, Ph.D., Science Buddies
Cite This Page
General citation information is provided here. Be sure to check the formatting, including capitalization, for the method you are using and update your citation, as needed.MLA Style
APA Style
Last edit date: 20170728
Share your story with Science Buddies!
Yes, I Did This Project! Please log in (or create a free account) to let us know how things went.Ask an Expert
The Ask an Expert Forum is intended to be a place where students can go to find answers to science questions that they have been unable to find using other resources. If you have specific questions about your science fair project or science fair, our team of volunteer scientists can help. Our Experts won't do the work for you, but they will make suggestions, offer guidance, and help you troubleshoot.Ask an Expert
Related Links
If you like this project, you might enjoy exploring these related careers:
Physicist
Physicists have a big goal in mind—to understand the nature of the entire universe and everything in it! To reach that goal, they observe and measure natural events seen on Earth and in the universe, and then develop theories, using mathematics, to explain why those phenomena occur. Physicists take on the challenge of explaining events that happen on the grandest scale imaginable to those that happen at the level of the smallest atomic particles. Their theories are then applied to humanscale projects to bring people new technologies, like computers, lasers, and fusion energy. Read moreMechanical Engineer
Mechanical engineers are part of your everyday life, designing the spoon you used to eat your breakfast, your breakfast's packaging, the fliptop cap on your toothpaste tube, the zipper on your jacket, the car, bike, or bus you took to school, the chair you sat in, the door handle you grasped and the hinges it opened on, and the ballpoint pen you used to take your test. Virtually every object that you see around you has passed through the hands of a mechanical engineer. Consequently, their skills are in demand to design millions of different products in almost every type of industry. Read moreMechanical Engineering Technician
You use mechanical devices every day—to zip and snap your clothing, open doors, refrigerate and cook your food, get clean water, heat your home, play music, surf the Internet, travel around, and even to brush your teeth. Virtually every object that you see around has been mechanically engineered or designed at some point, requiring the skills of mechanical engineering technicians to create drawings of the product, or to build and test models of the product to find the best design. Read moreNews Feed on This Topic
Looking for more science fun?
Try one of our science activities for quick, anytime science explorations. The perfect thing to liven up a rainy day, school vacation, or moment of boredom.
Find an ActivityExplore Our Science Videos
DIY Toy Sailboat

Make A Tissue Paper Parachute  STEM Activity

Stretchy Balloons! Fun STEM Activity
