A mass hangs from a light

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Numerical and computational mathematics

Computational Physics, Ch Finding the best parameters: minimisation and fitting

Problem

A mass

m=2.0\,\mathrm{kg}

hangs from a light, ideal spring of stiffness

k=50\,\mathrm{N\,m^{-1}}

, with

x

the downward extension from the spring's natural length. Air resistance and the spring's own mass are ignored. The energy of the mass, taking the gravitational potential energy reference at

x=0

, is

E(x)=\tfrac12 kx^2 - mgx,

with

g=9.8\,\mathrm{m\,s^{-2}}

. (a) Find the exact equilibrium extension

x^*

by setting

\mathrm{d}E/\mathrm{d}x=0

. (b)

E(x)

is minimised numerically over the bracket

x\in[0,1]\,\mathrm{m}

(width

W_0=1.0\,\mathrm{m}

) by golden-section search, which after its two starting evaluations shrinks the bracket by a factor of

0.618

at every further function evaluation. How many of those further evaluations are needed to shrink the bracket to a width below

1.0\,\mathrm{mm}

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