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Math Lab 23

Purpose of Lab• Demonstrate that the force used to stretch a

spring is proportional to the amount it stretches. Find the constant of proportionality.

Math Principles

• Direct variation, scatter-plot, slope, measuring with a ruler, linear regression.

What It’s All About• The amount of force (F) needed to stretch a

spring varies directly as the amount of stretch (x). The equation that describes this is:

F =(const)x.

What It’s All About• In this lab the stretching force is provided by

the weight of nuts hung on the end of a spring (slinky). Since this weight is proportional to the number of nuts, the equation will be modified to the following where N is the number of nuts:

N = kx

What It’s All About• Marks will be made on the background (chalk

board or white board) as several nuts are successively placed on the hanger. Distances between the marks are measured for later analysis in which the constant of proportionality will be determined.

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• Mark the starting point: With the spring hung with no added weights, stabilize it and mark the bottom of the spring. All measurements will be referenced from this point. Make sure the student marking this and following positions has his eyes level with the bottom of the spring so as to accurately mark the true position.

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• Gently hang one nut on the hanger, stabilize the spring, mark the position, and label with “1”. Again, make sure eye level is at the bottom of the spring so as to place an accurate mark.

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• One at a time, add nuts (up to a total of four) to the hanger. Each time mark the position of the bottom of the spring and record the number of nuts beside each mark. The picture below shows the marks and labeling after all four nuts have been attached.

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• Measure the distances: Using a centimeter ruler, measure the distances of each position of the spring (1, 2, 3, & 4) from the zero mark as shown to the left.

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• Putting it all together: Make a scatter-plot of the points with a large black marker.

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• Finding the constant of proportionality: Determine the constant of proportionality (k, also the slope of the line) for this line as follows:

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• Finding the constant of proportionality: Determine the constant of proportionality (k, also the slope of the line) for this line as follows:

• Since the line goes through the origin, use (0, 0) as one point and (36.4, 4) as the other. The slope (and therefore k) is:

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• Finding the constant of proportionality: Determine the constant of proportionality (k, also the slope of the line) for this line as follows:

• Since the line goes through the origin, use (0, 0) as one point and (36.4, 4) as the other. The slope (and therefore k) is:

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• Finding the constant of proportionality: Determine the constant of proportionality (k, also the slope of the line) for this line as follows:

• Since the line goes through the origin, use (0, 0) as one point and (36.4, 4) as the other. The slope (and therefore k) is:

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• Finally, the complete linear function is:

F = .10989x