I own a mobile home. The ceiling joists are 2" x 2". Will these support a pot rack that weighs about 35#, plus the pots wqhen they are hung on it?

Probably not but there are many factors that affect this.

In regular house construction, the minimum size ceiling joist allowed is 2x6 - no matter what the span. The strength of 2x2's is so far below this minimum that it's hard to imagine the joists carrying any other load.

That said, if these 2" joists are metal or if they have an upwards bow built into them then they can carry more. So it's really hard to say for sure how much yours can carry. If you have some big, heavy pots to hang on that rack then you may get up towards 100 pounds. While one 2x2 will not carry this without bowing, if you can spread the load over two or three joists then it may work.

Other than checking with the mobile home manufacturer the only way to check will be to hang some weight there and see what happens.

Really asking for trouble trying to do this.