Classical Mechanics (Kibble & Berkshire) · Problem 6.18

Orbits in a central field — T.W.B. Kibble, F.H. Berkshire

Physics — Mechanics · constrained-motion-parabola

Problem

A smooth wire has the form of the parabola z=x2/2bz = x^2/2b, y=0y = 0, where bb is a positive constant. The wire is fixed with the axis OzOz pointing vertically upwards. A particle PP, which can slide freely on the wire, is performing oscillations with xx in the range −a≤x≤a-a \leq x \leq a. Show that the period τ\tau of these oscillations is given by τ=4(gb)1/2∫0a(b2+x2a2−x2)1/2dx.\tau = \frac{4}{(gb)^{1/2}}\int_0^a \left(\frac{b^2 + x^2}{a^2 - x^2}\right)^{1/2}dx. By making the substitution x=asin⁡ψx = a\sin\psi in the above integral, obtain a new formula for τ\tau. Use this formula to find a two-term approximation to τ\tau, valid when the ratio a/ba/b is small.

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