Cycloids

• A dust particle is stick into the surface of a car’s wheel of radius $R$. (This is a 2D problem!)
  1. If the ground is set at $y=0$, and at some point, the dust particle is at the origin of the coordinates, find the parametric equations of the particle’s trajectory and draw a graph. This is called a (simple) cycloid.
  2. Now assume that a point mass is constrained to move on the graph you just drew. Neglect all frictions and show that in the presence of a uniform, upward gravitational field $\textbf{g}=+g\hat{y}$, the point mass oscillates harmonically with a time period $T = 4\pi\sqrt{R/g}$
  3. At a height $y$ from the ground, the point mass’ speed will be proportional to $\sqrt{y}$. Define $n(y)\equiv1/\sqrt{y}$ and show that on the cycloid, $n(y)\sin \theta$ is constant; where $\cot\theta$ is the slope of the curve. Now compare this result with snell’s law of refraction in geometric optics and invoke Fermat’s principle to realise that the cycloid provides the minimum travel time between the origin and any other point on it for a freely sliding point mass.
  4. A simple pendulum with (massless) string length of $L=4R$ is hanging (upwards) from the origin. Show that the curve on which the pendulum travels is also a cycloid and use this to calculate the oscillation period.