A bead slides without friction along a straight wire

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Mechanics & Dynamical Systems · Calculus of change and accumulation

Mathematical Methods for Physics, Ch The derivative as a physical instrument

Problem

A bead slides without friction along a straight wire, and its potential energy

U

against position

x

has been plotted. Values read off the plotted curve at

1.0\,\mathrm{cm}

steps, each good to

\pm0.05\,\mathrm{mJ}

, are:

U=9.0,\,2.0,\,0.0,\,2.0,\,5.0,\,6.0,\,4.5,\,4.0,\,4.5\,\mathrm{mJ}

x=0,\,1,\,2,\,3,\,4,\,5,\,6,\,7,\,8\,\mathrm{cm}

. ((a)) From the changes of sign in the differences between neighbouring values, find, to within one step, the minima and maxima of

U

that these readings show, where the force

F=-\mathrm{d}U/\mathrm{d}x

vanishes. Why can the table not show that these are all of them? ((b)) Estimate

F

x=4.0\,\mathrm{cm}

by a central difference, in newtons, with its reading uncertainty, and say which way it pushes the bead. ((c)) At each minimum, take the one-step slopes just before and just after it and divide their change by the

1.0\,\mathrm{cm}

step. Give this estimate of the curvature in

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

, and say which minimum these estimates make the stiffer.

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