It isn't really a homework problem rather
If you search " After they rotated the device by 90 degrees, the arm which was sideways to the ether would now be going with-and-against the wind, while the other arm would switch from there-and-against to sideways. This would cause the light beams to switch roles, so that the previous winner would now come in second place by the same difference. If Michelson and Morley compared the results from the two orientations, they ought to find a total difference which was double the difference in a single run: " on the link you will find where it talks about rotating the apparatus 90 degrees.
The Attempt at a Solution
I am having trouble understanding why the time delay when the apparatus is rotated 90 degrees is twice that of the time delay of the original setup. I've looked through various books like kleppner but found no helpful explanation. Is the time delay measuring the time as he is rotating the apparatus and light is traveling or is it after the rotates the apparatus. He sends out a beam of light and calculates the time delay.
I would think that the time would be the same if all there doing is switching roles.