A small bead of mass slides without friction inside a shallow bowl whose height above…

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

Modes, Packets and Media, Ch The anharmonic oscillator: frequency, harmonics and drift

Problem

A small bead of mass

m

slides without friction inside a shallow bowl whose height above its lowest point is

y(x) = \dfrac{x^2}{2R} + \dfrac{x^4}{4R^3}

, where

R

is a constant length and

x

is the horizontal distance from the lowest point. Use the simplified model that ignores the bead's vertical speed: the potential energy is

V(x) = mgy(x)

and the equation of motion is

m\ddot x = -\mathrm{d}V/\mathrm{d}x

. Take

R = 0.20\,\text{m}

and

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

. ((a)) Expand

V(x)

to fourth order in

x

and write the equation of motion in the standard form

\ddot x + \omega_0^2x + \eta x^2 + \varepsilon x^3 = 0

. ((b)) Give numerical values of

\omega_0^2

\eta

and

\varepsilon