1) Are you sure you are using a Firewire 400 port. The 24" has a FireWire 800 port which need different cables and a single Firewire 400 port...
2) Are you trying to use a 4 pin cable? You need a 6 pin end for the iMac
If you really want a new cable you need a 6 pin Firewire 400 cable. You may need a 6 pin to 6 pin cable or you may need a 6 pin to 4 pin cable: it depends on what's on the other end...