You can't copy the Video TS file you get from the ripping process directly to the iPhone. You must first encode the file into an MP4/M4V. Handbrake, for Mac, is by far extremely easy to use. The Windows version is straight forward, as well.
When you start up Handbrake (current version 0.92), select the iPhone preset; select the folder containing the ripped file as the source; select the path and file name for the output (MP4/M4V) file; select the iPod atom box; select a video bit rate that meets your file size needs (~1500 if you want to play on TV; ~500 if you are only going to play on the iPhone).