A frictionless slide is constructed in the shape of a cycloid. The horizontal coordinate x and vertical coordinate y of the slide are given in parametric form by

where \(A\) is a constant. Here the \(y\) coordinate is positive upward. The slide is the portion of the cycloid with \(=-\pi \leq \varphi \leq \pi\), with the bottom of the slide corresponding to \(\varphi=0\). There is a uniform gravitational field \(g\) in the negative \(y\) direction.

(a) Find the Lagrangian of a small block of mass \(m\) moving along the slide, using \(\varphi\) as the generalized coordinate.

(b) The block will oscillate back and forth near the bottom of the slide. Is its motion simple harmonic in the limit of small amplitudes? If not, explain why not; if so, find the angular frequency of oscillation \(\omega\) in terms of any or all of \(A, m\), and \(g\).

Transcribed Image Text:

x = A(+sin), y = A(1 - cos)