A small-amplitude pendulum of initial length swings with

PHYSICS · Multi-part · Fermi University Physics · Mechanics & Dynamical Systems · Oscillations and normal modes · Hamiltonian mechanics and canonical structure

Action and Symmetry, Ch Adiabatic invariants, canonical perturbation theory and integrable systems

Problem

A small-amplitude pendulum of initial length

\ell_0 = 1.00\,\mathrm{m}

swings with

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

. Its string is drawn up through the pivot so that its length falls linearly in time from

\ell_0

\ell_0/2

, the shortening completed after a time equal to

N

periods of the initial pendulum,

\tau_0 = 2\pi\sqrt{\ell_0/g}

. The pull exerts no torque about the pivot, so the small-angle motion obeys

\frac{\mathrm{d}}{\mathrm{d}t}\!\left(\ell^2\dot\theta\right) = -g\ell\theta.

Starting from

\theta(0) = 0.10

rad and

\dot\theta(0) = 0

, integrate this numerically for

N = 1

N = 10

and

N = 100

, and plot

E(t)/\omega(t)

against

t

in each case, where

E = \tfrac12m\ell^2\dot\theta^2 + \tfrac12mg\ell\theta^2

is the energy of the swing and

\omega(t) = \sqrt{g/\ell(t)}

. ((a)) Compute

\tau_0

. ((b)) The slowness ratio at the start of the ramp is

|\dot\ell/\ell|/\omega

, evaluated at

t=0

. Show that it equals

1/(4\pi N)

and evaluate it for

N = 1

10

100

. ((c)) State which of the three plots of

E/\omega

is expected to be flattest, and why. ((d)) From your integration, report the final value of

(E/\omega)/(E_0/\omega_0)

for each

N

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