Consider a single loop of the cycloid with a fixed value of a, as shown in figur
ID: 2173476 • Letter: C
Question
Consider a single loop of the cycloid with a fixed value of a, as shown in figure 6.11. A car is released from rest at a point Po anywhere on the track between O and the lowest point P (that is, Po has a parameter 0 < ?o < ?). Show that the time for the cart to roll from Po to P is given by the integraltime (Po --> P) = ?(a/g) integral(from ?_0 to ?) ?(1-cos?)/?(cos?0 - cos?) d?
and prove that this time is equal to ??(a/g). Since this is independent of the position of Po, the cart takes the same time to roll from Po to P, whether Po is at O, or anywhere between O and P, even infinitesimally close to P. Explain qualitatively how this surprising result can possibly be true.
Note: the intergrate is from ?o to ?, not ? to ?, I have proven that the time equals the above
integral but I can't figure out how to evaluate the integral. The book suggests using
the substitution ?=?-2u and converting the cosines to sines but I don't see how this makes the
integral any simpler.
Explanation / Answer
unfortunately you forgot to upload image.