Question

We want to determine the constant of a spring using Hooke's Law as reference. From the experiment we can extract the following pairs (Applied Force (N), How much the spring elogated (m)): {(2,4),(4,7),(7,15),(8,16)}.

a) Determine the value that minimizes the cuadratic error for the constant of the spring (F=kx)

b) Could we find an approximate solution that generates a residual vector r with cuadratic norm ||r||^2 = 0.45?(Remember that r = b - ?? ̂. )

Answer 1

(a) Plot the given data. Perform a least-square fit. Then the value of 'k' you will obtain will have minimum quadratic error.

I have plotted the given data in Excel. And the value of 'k' obtained from it (least-square fitting) is

k = 0.5 N/m (see the graph attached below).

(b) following are the residuals for all four data points

r = F (data) - K (estimated) * x (data)

r1 = 2 - 0.5*4 = 0 , ||r1||^2 = 0

r2 = 4 - 0.5*7 = 0.5 , ||r2||^2 = 0.25

r3 = 7 - 0.5*15 = -0.5 , ||r3||^2 = 0.25

r4 = 8 - 0.5*16 = 0 , ||r1||^2 = 0.