Potential Curvature and Nonlinear Oscillations
If U has a stable minimum at x0, then U is approximately U(x0) + U''(x0)(x-x0)^2/2 and omega^2 = U''(x0)/m; higher terms govern nonlinear corrections.
Why this shows up in the exam
Finding small-oscillation periods from U(x) · Amplitude scaling in power-law potentials · Piecewise or anharmonic oscillator periods
Learn the idea
Small oscillations depend on potential curvature, while nonlinear potentials can make period amplitude-dependent. Near a smooth minimum, almost any potential looks like a parabola; farther away its nonparabolic shape changes the timing and may destroy exact SHM.
🧠 Memory hook: At a minimum, curvature plays the role of a spring.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- k_eff = U''(x0) — local stiffness at a stable equilibrium
- omega² = U''(x0)/m — small-oscillation angular frequency
- T = sqrt(8m) integral₀^A dx/sqrt(E-U(x)) — period from energy for a symmetric one-dimensional potential
How to approach it
- 1Find the equilibrium from U prime equals zero
- 2Evaluate the second derivative there
- 3Retain nonlinear energy methods when no quadratic approximation is allowed
Common slip-ups that cost marks
- •Expanding about a non-equilibrium point
- •Using a negative curvature as a real frequency
- •Applying the harmonic period at large amplitude without justification
🌟 That's the whole idea — you've got this. Try the practice set below; every question you attempt makes it stick a little harder.
Original chapter practice
Original questions for this chapter, not past-paper questions or an exact mapping to this individual concept.
A mass of 1 kg is attached to a spring of force constant 100 N/m. Find the angular frequency of small oscillations.
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