A mass is attached to a spring that exerts a restoring force given by F(x) = -k x + x^ 3

PHYSICS · Multi-part · A Level · Physics · Further mechanics and thermal physics · Further mechanics and thermal physics

Problem

A mass

m

is attached to a spring that exerts a restoring force given by

F(x) = -k x + \alpha x^{3},

where

k

and

\alpha

are positive constants and

x

is the displacement from equilibrium. Assume motion occurs on a frictionless surface and is confined to one dimension. (a) Using qualitative reasoning, describe how the presence of the

\alpha x^{3}

term alters the restoring force at large displacements compared with an ideal spring obeying Hooke’s law only. (b) Demonstrate that for sufficiently small amplitudes the motion can be treated as SHM, and state the mathematical condition that the amplitude

A

must satisfy for this treatment to be valid. (c) For the small-amplitude regime identified in part (b), derive an expression for the angular frequency of oscillation in terms of

k

and

m

. (d) Suppose the amplitude is increased so that the

\alpha x^{3}

term is no longer negligible. Predict qualitatively whether the period becomes longer, shorter, or remains unchanged compared with the small-amplitude period, and justify your prediction using energy or force considerations.

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