Applying Hooke's Law: Make Your Own Spring Scale

Abstract

Hooke's law says that the opposing force of a spring is directly proportional to the amount by which the spring is stretched. How accurately Hooke's law describe the behavior of real springs? Can springs be used to make accurate scales for weighing objects? Spring into action and find out for yourself with this project.

Objective

The goal of this project is to investigate Hooke's law and see how a spring can be used to weigh objects.

Credits

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Last edit date: 2020-11-20

Introduction

Under some conditions, a spring has an interesting property that was discovered by the physicist Robert Hooke. The property is described by an equation now known as Hooke's law. Hooke's law says that the restoring force (F) produced by the spring is proportional to the distance by which the spring has been lengthened (x). In equation form, Hooke's law looks like Equation 1:

$$F = -kx$$

• F is the restoring force in newtons (N)
• k is the spring constant in newtons per meter (N/m)
• x is the distance the spring has been stretched (or compressed) from its neutral length in meters (m)

The equation says that the force (F) of the spring is equal to the spring constant (k, a measure of the stiffness of the spring) times the distance (x) that the spring has been stretched. The minus sign says that the force is exerted in the opposite direction of the stretching. In other words, if you stretch the spring out, the spring force is pulling back in the other direction.

As anyone who has stretched a Slinky® a bit too much knows, if you pull the spring too far, Hooke's law no longer applies. The part of the spring that is stretched too much doesn't spring back any more, because the stretching went beyond the elastic limit of the spring material. When this happens, the spring usually ends up with a visible kink where the excessive stretching occurred. So there are certainly some conditions where Hooke's law does not apply.

This experiment is to test whether Hooke's law accurately describes the stretching of a spring over some range. Can you calibrate a spring and then use it to weigh objects of unknown mass? Try it for yourself and find out.

Terms and Concepts

• Spring
• Hooke's law
• Restoring force
• Spring constant
• Elastic limit

Questions

• How can a spring be used to weigh an object? Can you find real-life examples of spring scales?
• What happens when a spring is extended beyond its elastic limit?

Materials and Equipment

To do this experiment you will need the following materials and equipment:

• Several springs with different lengths, diameters, and stiffness. Springs are available at hardware stores, but you can also find them by disassembling click pens or some toys.
• Weights to hang from springs, here are some tips:
  • You can use metal weights, or hang a container from the spring and fill it with coins, water, or sand. For the container, you can use a small bucket, or make a "bucket" by poking holes toward the top of a paper or plastic cup and attaching string as a handle (do not use for heavy weights).
  • The appropriate weight to be used with each spring depends on the stiffness of spring, you will have to use your judgment to decide how much weight to use with each spring. Experiment with the springs by pulling on them with your hands, to get an idea for how stiff they are.
• Sturdy support from which to hang the springs. These materials are available at a hardware store. For example:
  • Wood
  • Sturdy cup hook
  • Clamp
  • Screw the cup hook into one end of the piece of wood, clamp the other end of the wood to a table or workbench, then hang the spring from the cup hook. The clamp and wood will both need to be strong enough to support the weights you intend to hang from your springs
• Kitchen scale for measuring actual mass of weights used
• Metric ruler

Experimental Procedure

1. Do your background research so that you are knowledgeable about the terms, concepts, and questions in the Background section.
2. For each spring, do the following steps. See Figure 1 for an example experimental setup.
   1. Hang the spring from a sturdily-mounted hook.
   2. Measure the length of the spring with no weight hanging from it. Always measure between the same two points on the spring (you may even want to mark them).
   3. Hang a weight from the spring, and wait for it to settle.
   4. Again measure the length of the spring.
   5. Measure the mass of the weight, using the kitchen scale. Important: make sure you include the mass of any supporting structures or containers, like the paper clips and bucket in Figure 1, in your measurement.
   6. Repeat for a series of different weights.
   7. Remove the weight from the spring, and check to make sure that the spring returns to its initial length. If the spring does not return to its initial length, you have added too much weight and stretched the spring past its elastic limit. You can keep doing your experiment, but make sure you take note of this in your lab notebook.
   8. Do at least three trials for each spring.
   
   Figure 1.  An example experimental setup.  A cup hook is screwed into a scrap piece of wood, which is clamped to the side of a workbench. The spring is hung from the cup hook, and a plastic bucket is hung from the spring using paper clips. The bucket can be filled with weights (sand, coins, water etcetera) to stretch the spring.
   
   
3. Keep track of your results in tables like Table 1. You will need a new table for each spring.

4. To calculate the change in length, subtract the average length of the spring with no weight (0 g) from the averaged measured length for each of the other weights.
5. For each spring, make a graph of the change in length of the spring (in cm, y-axis) vs. the mass of the weight hanging from the spring (in g, x-axis).
6. More advanced students should graph the change in length of the spring (in cm, y-axis) vs. the force on the spring (in newtons, x-axis). The force on the spring in newtons can be obtained by multiplying the mass (in kg) by the acceleration due to gravity: 9.8 m/s². Remember that to convert g to kg, you have to divide by 1000.
7. From your graph, does it appear that your spring is following Hooke's Law?
8. What is the spring constant (k) for each spring? Be careful when you calculate this. The spring constant is the slope of a graph of force vs. displacement, but that is not what you plotted in step 5.
9. With the help of the matching graph, can you use your spring to measure the weight of another object? Hang an object of unknown weight from the spring, then measure the spring's change in length. Use your graph to determine how much the object should weigh. Check how accurate your measurement is by weighing the object on the kitchen scale.

If you like this project, you might enjoy exploring these related careers:

Variations

• For a more advanced experiment using a spring-based mechanical model of the human knee, see the Science Buddies project Deep Knee Bends: Measuring Knee Stress with a Mechanical Model.
• Read about nonlinear springs in the Linear & Nonlinear Springs Tutorial. Does that information apply to your results?
• Repeat the experiment with other "stretchy" objects like rubber bands. How do your results differ?
• Can you figure out a way to do the experiment by squishing the springs (compression) instead of stretching them (tension)?

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