Galileo’s Paradox
Math 211  Prof. Leise

Email completed labs to tleise@amherst.edu by noon Monday.

Galileo’s paradox: The time for a bead to slide down a line segment starting at a point on a circle to the bottom of the circle is always the same, no matter where on the circle the line segment begins.  Recall (from the Brachistochrone lab) that the traversal time is given by a line integral with respect to arc length:

(∫_C)(1/(Sqrt[2g((Subscript[y, 0])-y)]))ds

where (Subscript[y, 0]) is the initial y-coordinate.  

Assuming that the circle has radius R, find a parametrization for the line segment and compute the traversal time.  You should find that the traversal time doesn’t depend on the starting point on the circle, that is, it takes the same amount of time for the bead to slide down the line segment no matter where the starting point on the circle is.  You can either type your work into Mathematica or write it out on a piece of paper to turn in.