We discussed (will discuss) the cycloid - the curve that traces the path travelled by a fixed point on a rolling circle, parameterised by the rolled angle 8. It turns out that the cycloid has some very striking properties. In this Problem, you will investigate these properties. Consider a uniform cylinder of mass M and radius R smoothly rolling down a curve with negligible friction under near-surface constant gravity. The cylinder starts from rest at some arbitrary point on the curve, and it rolls down to some other point that is not vertically below it. We shall neglect air resistance. 

(a) As with rolling problems, let us first consider the translational part of the rolling. The translational motion is described with the centre of mass of our cylinder, as a point mass of M. A big-brain Italian bloke from the 1500's once guessed that a circular curve would minimise the time it took for our point mass. Was he correct or incorrect? Hint: Start by finding a condition/equation that would identify such a minimal-time curve for our point mass. 

(b) Later in the late 1600's, multiple big-brain blokes across Western Europe claimed that the cycloid is the correct curve that minimises the time for our point mass. Were they correct or incorrect? 

(C) We now return to the cylinder to account for the rotational part of its rolling. Here, it is our turn to now claim something - let us claim that a curve parallel to the cycloid is the minimal-time curve for our rolling cylinder. Is our claim correct or incorrect, and what is said curve that is parallel to the cycloid that minimises the time taken by our rolling cylinder? 

(d) Consider the time taken by the cylinder rolling from rest at an arbitrary initial point to the bottom of the minimal-time curve. Does this time depend on the initial point of our cylinder? What about the time for the curve for the centre of mass as a point mass?

 (e) Based on our answers, what is an engineering application we can consider for the cycloid curve?