Actually, it has to do with bullet length and shape, not weight. For example, a VLD (Very Low Drag) bullet made of solid copper will be lighter and longer than a round nosed bullet with a lead core and copper jacket but the round nosed bullet will not need as tight a twist to stabilize. This is because the aerodynamics of a long pointy bullet make it inherently more unstable in flight so it needs more gyroscopic force to keep it point on to it's target in flight.
kyhunter gives a pretty good summary, however. The lighter bullets will do fine in a 1:9 twist or even a 1:7 twist so long as the jacket is tough enough to withstand the RPMs. For a modern 5.56/.223, a 1:9 twist will work well with most 55 gr bullets and sometimes with some of the longer ones, especially if it's a 20" barrel. For carbines, because the shorter barrel generally produces less velocity (thus imparting a lower RPM to the bullet) a 1:7 or 1:8 is a better choice.
The caveat is, every barrel is a law unto itself. You can take two barrels that came down the same assembly line, one right after the other, made to identical specs and have one that will shoot longer bullets well, the will not and vice versa