# 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**

x_{i}-mean height, y_{i}-force of gravity

** (1) Mean Height (cm):** 10^{th} cell

(Height_{up} + Height_{down})/2 = (35.4+35.6)/2 = **35.5 cm**

** (2)** **Force _{g} (N): **2

^{nd}cell

F_{g} = ma = 0.150(kg ) x 9.81(m/s^{2}) = **1.47 (N)**

** (3) x _{i}y_{i} (cJ):** 2

^{nd}cell

x_{i}y_{i }= 74.5(cm) x 1.47(N) = **110 (cJ)**

** (4) (x _{i})^{2} (cm^{2}):** 1

^{st}cell

(x_{i})^{2} = 77.3(cm) x 77.3(cm) = **6000 (cm ^{2})**

** (5) f(x _{i}) (N): **3

^{rd}cell

f(x_{i}) = f(0.250) = -0.202(0.250) + 16.5 = **2.50 (N)**

** (6) [y _{i}-(f(x_{i})]^{2} (N): **3

^{rd}cell

[y_{i}-(f(x_{i})]^{2} = [2.45-2.50]^{2} =** 0.210 (N)**

** (7) [y _{i}-y_{bar}]^{2 }(N): **10

^{th}cell

[y_{i}-y_{bar}]^{2} = (y_{10}-[SUM(y_{1}:y_{11})/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 r

^{2}values differed by 0.20%. The least squares method provided an r

^{2 }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.