A spring's extension is measured for nine equally spaced applied loads

PHYSICS · Multi-part · Fermi University Physics · Scientific Practice & Experimental Methods · Designing and running experiments · Designing and running experiments

Measurement and Method, Ch Designing the measurement: noise, averaging, null and differential methods

Problem

A spring's extension xx is measured for nine equally spaced applied loads FF, with the same standard uncertainty u(x)u(x) on every reading and the load values known exactly. The extension is fitted to x=mF+c0x = mF + c_0, so the slope gives m=1/km = 1/k, the inverse spring constant. For NN equally weighted points, the standard uncertainty of the fitted slope isu(m)=u(x)∑i(Fi−Fˉ)2.u(m) = \frac{u(x)}{\sqrt{\sum_i (F_i - \bar F)^2}}. ((a)) Nine equally spaced loads span a total range WW, running from Fˉ−W/2\bar F - W/2 to Fˉ+W/2\bar F + W/2. Show that ∑i(Fi−Fˉ)2=0.9375 W2\sum_i (F_i - \bar F)^2 = 0.9375\,W^2. ((b)) Experiment 1 uses nine equally spaced loads from 1.0 N1.0\,\mathrm{N} to 9.0 N9.0\,\mathrm{N} (spacing 1.0 N1.0\,\mathrm{N}). Experiment 2 uses nine equally spaced loads from 4.0 N4.0\,\mathrm{N} to 6.0 N6.0\,\mathrm{N} (spacing 0.25 N0.25\,\mathrm{N}). Both use the same u(x)u(x) on every reading. By what factor is u(m)u(m) smaller in experiment 1 than in experiment 2?

Browse the problem library