(a) A block of mass on a horizontal spring of stiffness oscillates with amplitude .…

PHYSICS · Subjective · Fermi University Physics · Mechanics & Dynamical Systems · Plasma Physics · The plasma state and single-particle motion

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

Problem

(a) A block of mass

0.50\,\mathrm{kg}

on a horizontal spring of stiffness

k_0 = 200\,\mathrm{N\,m^{-1}}

oscillates with amplitude

A_0 = 0.10\,\mathrm{m}

. The stiffness is raised slowly (many periods) to

k_1 = 800\,\mathrm{N\,m^{-1}}

, with the mass otherwise undisturbed. Find the new amplitude

A_1

. (b) A charged particle gyrates about a field line with pitch angle

\alpha_0 = 20^\circ

where the field is

B_0 = 0.050\,\mathrm{T}

. As it drifts into a region of slowly increasing field it is turned back (mirrored) at some field

B_{\rm turn}

. Find

B_{\rm turn}

. (c) A one-degree-of-freedom system has

H = H_0(I) + \epsilon H_1(I,\vartheta)

with

H_0 = \omega_0 I

\omega_0 = 50\,\mathrm{rad\,s^{-1}}

, and

\epsilon H_1 = 4.0\,I^2\sin^2\vartheta

(SI units throughout). For an action

I = 0.020\,\mathrm{J\,s}

, find the shifted angular frequency

\nu'