A thin steel strip clamped at both ends and pushed into a bow has two stable shapes

PHYSICS · Multi-part · Fermi University Physics · Mechanics & Dynamical Systems · Oscillations, Waves & Acoustics · Simple, damped and driven oscillations

Mathematical Methods for Physics, Ch Second-order linear equations: damping, driving and resonance

Problem

A thin steel strip clamped at both ends and pushed into a bow has two stable shapes, bowed up and bowed down. The displacement

x

of its midpoint is modelled by the potential

U(x) = U_0\big[(x/a)^2-1\big]^2

, with

U_0 = 0.020\,\mathrm{J}

and

a = 1.0\,\mathrm{cm}

, and the moving part has effective mass

m = 0.010\,\mathrm{kg}

. ((a)) Find where

U'(x)=0

, and use the sign of

U''

to say which of these points are minima. ((b)) Find the small-oscillation angular frequency

\omega_0

about

x=a

. ((c)) Writing

x=a+s

, expand

U

in powers of

s

and find the displacement

s

at which the cubic term reaches one tenth of the quadratic one. ((d)) Explain why there is no small-oscillation frequency about

x=0

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