Academic journal article Science Scope

# Springing into Linear Models

Academic journal article Science Scope

# Springing into Linear Models

## Article excerpt

In eighth grade, students usually learn about forces in science class and linear relationships in math class, crucial topics that form the foundation for further study in science and engineering. An activity that links these two fundamental concepts involves measuring the distance a spring stretches as a function of how much weight is suspended from the spring. After students have collected data, they plot the data on graph paper and determine that the relationship is linear. Students then calculate the slope of a best-fit line and construct a linear model. This model is used by students to predict how far an additional weight will stretch the spring. Finally, students test their prediction and examine how accurate their model is. This activity requires two 40-minute classes to complete and should be conducted toward the end of a unit on force and motion. Because this activity is interdisciplinary and student centered, and yields excellent (reproducible) results, students remain engaged and interested as they learn about how math is used in the "real world" of science and engineering.

Background for teachers

The first person to discover the linear relationship between the force pulling on a spring and the distance the spring extends was an English physicist, Robert Hooke, and this linear relationship is usually called Hooke's law (Giancoli 2004). In textbooks, Hooke's law is usually stated as F = kx, where F is the force required to extend a spring a distance, x, and k is the constant of proportionality (Giancoli 2004). From the small springs inside pens, to the larger springs inside shock absorbers in cars, to extremely large springs used to build earthquake-resistant structures, engineers and scientists use Hooke's law to model spring behavior. Because of this, most vendors refer to the experimental setup used in this activity as a "Hooke's law" apparatus (shown in Figure 1). This apparatus can be purchased for \$15-\$25 from major online retailers (see Resources). If you are going to have students work in pairs, which I recommend, it means an initial expense of about \$200 for a typical class size. These setups are quite durable and last for many years. If resources are limited, it may be possible to borrow setups from a high school or local university. Writing a grant is also an option; PTAs and other organizations routinely provide funds for science equipment. Even if only one Hooke's-law apparatus can be purchased, consider performing this experiment as a class activity. Every student will get the opportunity to make scientific measurements, and then data from the entire class can be analyzed.

The initial assembly of the apparatus is very straightforward, taking less than five minutes to complete, with complete instructions included from the supplier. A support rod is screwed into a sturdy base. Then a 15 cm vertical scale is attached to the support rod, which can be easily adjusted up or down on the support rod. A hook is secured to the top of the support rod, from which a spring is suspended. A lightweight plastic hanger is attached to the bottom of the spring. The apparatus comes with 10 0.1 N weights that can be placed on the hanger, stretching it a certain distance depending on how many weights are used. There is a horizontal wire called the distance indicator attached to the hanger, which measures how far the spring has stretched. It's important to move the adjustable scale up or down the support rod until the scale reads 0 when there are no weights on the hanger. The fact that the spring stretches zero distance when zero weight is applied to it means that the relationship is not just linear but a direct proportionality (directly proportional quantities are linear, with the line going through the origin).

Once assembled, the apparatus requires no maintenance and, as shown in Figure 1, it's quite straightforward to use: A student suspends one of the 0.1 N weights from the spring, which stretches the spring, and the distance indicator measures how far the spring has stretched due to this 0. …

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