Why America uses 60 Hz as operating frequency in power systems & electric machines & residential appliances while the rest of the world uses 50 Hz??

The present frequency use is the end result of the "battle of the Hertz" in the early part of the 1900's. There were very many other frequencies used (including such strange ones as 16.67 and 25 -- mostly driven by traction applications) but these were the ultimate final ones settled on. Note there are other parts of the world where 60 Hz is also the standard (I think portions of Japan besides areas in South America).

A good source of the history is to check out the IEEE Power Engineering Society. They had a series of historical articles over the last several years in their bi-monthly magazine. I am not certain if they are online in a public site, but your school library staff may be able to assist in getting access.

Frequency 16 2/3 Hz of electric traction systems origins from machine conversion 1 pair poles to 3 pair poles... There was 42 Hz frequency in Italy 100 years ago...