# Linear fitting with Hooke’s Law

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Assuming that this is the data for a linear experiment with ideally, zero y-intercept:

### Data

Click here to download the data or copy the following data into a notepad and save it as a text file. I normally prefer to change the filetype from “.txt” to “.dat”, simply because, it contains data.

 # m (g) Data 1 Data 2 Data 3 average std.dev 200 5.683631 6.29861 8.381023 6.787755 1.413658 400 10.23901 13.5709 15.18523 12.99838 2.522326 600 11.84209 11.36025 9.422947 10.8751 1.280467 800 16.33589 16.92676 19.82742 17.69669 1.868766 1000 22.32341 23.72259 20.2721 22.10603 1.735483 1200 24.86626 27.70103 28.07837 26.88189 1.755747 1400 28.19161 29.78272 31.26129 29.74521 1.535185 1600 34.77228 34.02847 30.23366 33.01147 2.434233 1800 37.84083 38.05711 36.07914 37.32569 1.084949 2000 37.64023 38.79702 42.7417 39.72632 2.674686

In this data, the first column is the mass (in grams), hung under a spring.
The spring extends and the elongation is recorded.
The experiment is repeated three times to produce three sets of data.
The average and standard deviation was calculated for these data.

For this data, we would use the linear equation y = mx + c.
You would need to produce 5 fits and plots to compare the results.

1. Fit and plot for Data #1 (not the average), where the parameters m and c are fitted (adjusted) to the graph.
2. Fit and plot for Data #1 (not the average), where only parameter m is fitted and parameter c is fixed to zero.
3. Fit and plot for Average Data, where the parameters m and c are fitted (adjusted) to the graph, without standard deviation data.
4. Fit and plot for Average Data, where the parameters m and c are fitted (adjusted) to the graph, with standard deviation data.
5. Fit and plot for Average Data, where only parameter m is fitted and parameter c is fixed to zero with standard deviation data.

### Required Knowledge

To perform the tasks about, one would require the basic knowledge of importing data into gnuplot.

y1(x) = m1*x + c1
m1 = 2
c1 = 0.1
fit y1(x) ‘data.dat’ using 2:5 via m1,c1
plot ‘data.dat’ using 2:5 with points
replot y1(x) with line t “y1(x) : Single Data with y-intercept”

Explanation

y1(x) = m1*x + c1

First, we would need to define the fitting equation.  In this case, the fitting equation is a simple linear equation with two variables, m1 and c1.

m1 = 2
c1 = 0.1

Then we assign a starting value for m1 and c1. Do not write zero.  Try to write something within the same order of the actual value.  The only way to know the actual value to to try to plot the graph without fitting.

fit y1(x) ‘data.dat’ using 2:5 via m1,c1

Next, this command fits the y1(x) equation with the data ‘data.dat‘.  The command “using 2:5” selects only columns 2 and 5 for data-fitting.  The values which are adjusted are written using “via m1,c1” where m1 and c1 will be changed to get the best fit.

plot ‘data.dat’ using 2:5 with points

This command plots the original data with data points

replot y1(x) with line title “y1(x) : Single Data with y-intercept”

This command plots the y1(x) equation that has been fitted with a title.

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