# Stretching Rubber Bands

We can use common household objects to measure properties that match physical laws. This experiment takes a very common household item, the rubber band, and applies physical laws (Hooke’s Law and the Young’s Modulus) to them in a hands-on way.

Purpose:

To describe the stretching action of rubber bands, and explore the connection between Hooke’s Law and Young’s modulus.

Introduction:

Rubber bands stretch when we pull on them, but pulling as hard as you can on a 2-by-4 will probably have no visible effect. The stretchability of solid materials is expressed as their Young’s Modulus (a.k.a. “Elastic Constant”), $Y$. Here is the formula for Young’s modulus (Eqn.1):

$Y=\dfrac{\dfrac{F}{A}}{\dfrac{\ \Delta L\ }{L_0}} \tag{1}$

• $F$ = Force applied to solid [N]
• $A$ = Cross-sectional area of solid [m$^2$]
• $L$ = stretched length of solid [m]
• $L_0$ = original length of solid [m]

A simple way to understand this formula is $Y = \frac{\text{stress}}{\text{strain}}$. The stress is the amount of force applied to the object, per unit area ($F/A$). The strain is the relative change in the length of the solid ($\Delta L/L_0$). Therefore, a solid with a greater value of $Y$ will stretch less than a solid with a smaller $Y$, when the same force is applied.

Let’s return to rubber bands. Rubber bands are elastic solids and can be described with Hooke’s Law (Eqn.2). We can think of Hooke’s Law as a simplified version of Young’s Modulus, and it is classically applied to spring systems. However, it can also, to some extent, describe the stretch patterns observed for rubber bands.

$F=k \Delta L \tag{2}$

• $F$ = Force applied to elastic material [N]
• $k$ = spring constant [N/m]
• $ΔL$ = change in length of the elastic material [m]

If you compare the two equations, you will find (try this as an exercise) that the spring constant $k$ contains Young’s modulus $Y$ (which describes the material), the length $L_0$, and the cross-sectional area $A$ of the material, can be related as in Eqn.3.

$k=Y\dfrac{A}{L_0} \tag{3}$

This allows us now to make predictions before we do an experiment. For example, a thicker rubber band should have a larger spring constant due to its larger cross-sectional area. In this experiment you can check this prediction and investigate the way in which Hooke’s Law applies to rubber bands. You can also think about what happens if you use two rubber bands at the same time, either to hang an object from both bands in parallel or to create a longer band by knotting one band to the end of the other band. Write down your hypothesis and test it with an experiment.

The Challenge:

Design an experiment to measure the constant $k$ for rubber bands. Use items of known mass to provide the applied force. Measure the change in length and the original length for each rubber band; also record the physical properties of each band.

Key Concepts:
• Young’s modulus is a measure of stress over strain.
• Hooke’s Law takes only applied force and change in length into account.
• Different rubber bands will have different constants for both laws.

Skills:
• Applying Hooke’s Law
• Relating graphs of experimental data to given equations
• Understanding relationship between Hooke’s Law and Young’s modulus
• Simple graphical analysis
• Assigning errors and understanding error calculations

Materials/Equipment:
• Three rubber bands of different sizes and thicknesses
• Objects of given weight (granola bars, packaged foods, etc.)
• Small metal hanger
• Pushpin
• Ruler (30cm) or flexible tape measure

Suggested assigned time: 2 weeks

• Why does Hooke’s law not apply for greater forces?
• Why is Young’s modulus a more general descriptor of rubber band action than Hooke’s law?

Variations:
• Try the experiment with something other than a rubber band.
• Compare rubber band action with spring action. How do the graphs for Hooke’s law compare?
• Combine multiple rubbers bands and analyze stretching action.