Lab Parts: Evaluating Spring Constants with the Least Squares Method of Data Fitting

Student Researched Lab Experiment to evaluate Spring Constants

Introduction

In this experiment we evaluated Hooke’s law by measuring the stretch of a spring with incremental masses. A motion detector provided position measurements and the masses were added 100 grams at a time. To calculate the spring constant we utilized the least squares method of data fitting on a Force by distance graph.

Samples Calculations

Line of Best Fit: f(x) = 0.202x + 16.5

xi-mean height, yi-force of gravity

(1) Mean Height (cm): 10th cell

(Heightup + Heightdown)/2 = (35.4+35.6)/2 = 35.5 cm

(2) Forceg (N): 2nd cell

Fg = ma = 0.150(kg ) x 9.81(m/s2) = 1.47 (N)

(3) xiyi (cJ): 2nd cell

xiyi = 74.5(cm) x 1.47(N) = 110 (cJ)

(4) (xi)2 (cm2): 1st cell

(xi)2 = 77.3(cm) x 77.3(cm) = 6000 (cm2)

(5) f(xi) (N): 3rd cell

f(xi) = f(0.250) = -0.202(0.250) + 16.5 = 2.50 (N)

(6) [yi-(f(xi)]2 (N): 3rd cell

[yi-(f(xi)]2 = [2.45-2.50]2 = 0.210 (N)

(7) [yi-ybar]2 (N): 10th cell

[yi-ybar]2 = (y10-[SUM(y1:y11)/11])2 = (10.30-5.40) = 24.1 (N)

Conclusion
The experiment was highly successful with a 1.00% discrepancy for the experimental spring constant. Data was also plotted on an excel graph and the r2 values differed by 0.20%. The least squares method provided an r2 value of 0.9978. It seems the experimental spring constant is highly accurate and perhaps a better representation than the accepted value of 0.200.