A light spiral spring hangs vertically from a fixed clamp

PHYSICS · Multi-part · Fermi University Physics · Scientific Practice & Experimental Methods · Uncertainty in measurement · Data analysis and model fitting in the laboratory

Measurement and Method, Ch Graphs, straight lines and least squares

Problem

A light spiral spring hangs vertically from a fixed clamp. Masses are hung from its lower end to apply six known forces

F

, and the extension

x

of a pointer against a fixed millimetre scale is read for each. The spring obeys Hooke's law over this range and the clamp does not move. No separate uncertainty was recorded for each reading, so the scatter of the points about the fitted line must supply it. Data:

F=1.0,\ 2.0,\ 3.0,\ 4.0,\ 5.0,\ 6.0\ \mathrm{N}

with

x=10.1,\ 18.2,\ 26.0,\ 33.9,\ 42.1,\ 49.9\ \mathrm{mm}

respectively. ((a)) Find the least-squares slope

m

and intercept

c_0

x/\mathrm{mm}

plotted against

F/\mathrm{N}

. ((b)) Estimate the scatter

s_y

of the points about this line, using the appropriate divisor for a two-parameter fit, and use it to find the standard uncertainties

u(m)

and

u(c_0)

. ((c)) The spring constant is

k = 1/m

. Give

k

and its standard uncertainty

u(k)

\mathrm{N\,m^{-1}}

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