I need to cut these lines across the arc. I only need the little itty bitty line ends. The arc and long parts of the lines are to be discarded.
In order to use the circle to trim the line paths, you must select the circle and convert it to a path by clicking Path > Object to Path or Shift+Ctrl+C on Windows
Select all paths except the circle and go to Path > Combine or Ctrl+K ; this allows you to perform boolean operations on the group of lines.
Make sure the circle is above your lines with Object > Raise to Top. Then select all paths and go to Path > Cut path or Ctrl+Alt+/ This will cut the combined group along the perimeter of your circle.
Finally, you can select each line trimming and delete them.