Question

part 3- graphing (25 pts): plot spring force versus displacement for the following data. then, draw a best-fit line through the points, and calculate the slope. be sure to label your graph completely and correctly. show all work for your slope calculation.

spring force (n)	displacement (m)
3.3	0.15
4.2	0.20
6.1	0.30
9.3	0.45
10.2	0.50
12.1	0.60
14.1	0.70

Explanation:

Step1: Choose two points

Let's pick the first point \((x_1, y_1)=(0.10, 2.1)\) and the last point \((x_2, y_2)=(0.70, 14.1)\) from the data. These points are likely on the best - fit line (or close to it as we are looking for the slope of the best - fit line for the linear relationship between spring force and displacement, which follows Hooke's law \(F = kx\) where \(k\) is the slope).

Step2: Use the slope formula

The formula for the slope \(m\) between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(m=\frac{y_2 - y_1}{x_2 - x_1}\).

Substitute \(x_1 = 0.10\), \(y_1=2.1\), \(x_2 = 0.70\) and \(y_2 = 14.1\) into the formula:

\(m=\frac{14.1-2.1}{0.70 - 0.10}=\frac{12.0}{0.60}=20\)

We can also check with other points. For example, take \((0.20,4.2)\) and \((0.60,12.1)\). The slope would be \(\frac{12.1 - 4.2}{0.60 - 0.20}=\frac{7.9}{0.40}=19.75\approx20\), which is consistent with our first calculation.

Answer:

The slope of the best - fit line (which represents the spring constant \(k\)) is approximately \(20\space N/m\) (the value may vary slightly depending on the choice of points, but the approximate value is around \(20\space N/m\)).