In a laboratory experiment

MATHEMATICS · Multi-part · A Level · Mathematics · Data presentation and interpretation · Data presentation and interpretation

Problem

In a laboratory experiment, five measurements of force FF (in newtons) required to extend a spring by a displacement xx (in centimeters) are recorded below. (x,F)(x, F): (1.0, 2.8),  (2.0, 5.5),  (3.0, 8.1),  (4.0, 11.2),  (5.0, 14.0)(1.0,\,2.8),\;(2.0,\,5.5),\;(3.0,\,8.1),\;(4.0,\,11.2),\;(5.0,\,14.0) Assume the spring obeys an idealized linear model that passes through the origin, F=mxF=m x, where mm is the spring constant (in N/cm). The constant mm is chosen to minimize the total squared difference between the measured forces and the model predictions. (a) Construct an explicit expression E(m)E(m) for the total squared difference between the measured forces and the model F=mxF=m x. (b) Determine the value of mm that minimizes E(m)E(m). (c) Explain whether adding additional data points that also lie near the same line through the origin would change the analytic formula you obtained for mm in part (b). Justify your reasoning. (d) Interpret the numerical value of mm you found in part (b) within the context of Hooke’s Law.

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