A particle of mass 0.75 kg moves near x = 0 in U(x) = ½(26 N m⁻¹)x² + (51 J m⁻⁴)x⁴. Determine the small-amplitude angular frequency and period, and state why the quartic term can be neglected in this limit.

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Accepted Answer

ω = 5.89 rad s⁻¹; T = 1.07 s Near x = 0, the x² term dominates x⁴, so U ≈ ½kx² with k = 26 N m⁻¹. ω = √(k/m) = √(26/0.75) = 5.89 rad s⁻¹. T = 2π/ω = 1.07 s; the quartic contribution decreases faster than the quadratic contribution as x → 0.

Small oscillations in a potential well 7

A particle of mass 0.75 kg moves near x = 0 in U(x) = ½(26 N m⁻¹)x² + (51 J m⁻⁴)x⁴. Determine the small-amplitude angular frequency and period, and state why the quartic term can be neglected in this limit.

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