A small puck of mass sits at the bottom of one corrugation of a horizontal track…

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 puck of mass

m = 0.50\,\text{kg}

sits at the bottom of one corrugation of a horizontal track whose potential energy is

V(x) = V_0(1-\cos kx)

, with

V_0 = 8.0\times10^{-3}\,\text{J}

and

k = 10\,\text{m}^{-1}

x

the horizontal displacement from the bottom of the corrugation. Friction and air resistance are ignored. ((a)) Expand

V(x)

to fourth order in

x

. ((b)) Write the equation of motion in the standard form

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

and give numerical values of

\omega_0^2

\eta

and

\varepsilon