Classical Mechanics (Kibble & Berkshire) · Problem 6.19

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

Physics — Mechanics · constrained-motion-cycloid

Problem

A smooth wire has the form of the cycloid x=c(θ+sin⁡θ)x = c(\theta + \sin\theta), y=0y = 0, z=c(1−cos⁡θ)z = c(1 - \cos\theta), where cc is a positive constant and the parameter θ\theta lies in the range −π≤θ≤π-\pi \leq \theta \leq \pi. The wire is fixed with the axis OzOz pointing vertically upwards. [Make a sketch of the wire.] A particle can slide freely on the wire. Show that the energy conservation equation is (1+cos⁡θ)θ˙2+gc(1−cos⁡θ)=constant.(1 + \cos\theta)\dot{\theta}^2 + \frac{g}{c}(1 - \cos\theta) = \text{constant}. A new parameter uu is defined by u=sin⁡12θu = \sin\tfrac{1}{2}\theta. Show that, in terms of uu, the equation of motion for the particle is u¨+(g4c)u.\ddot{u} + \left(\frac{g}{4c}\right)u. Deduce that the particle performs oscillations with period 4π(c/g)1/24\pi(c/g)^{1/2}, independent of the amplitude!

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