Broken cannon cogs are used when repairing dwarf multicannons within the Artisans Workshop. You will receive 10 broken cogs after removing them from a broken cannon base. You should try to repair the broken cogs on an anvil with a hammer, to make fixed cogs. The repaired cogs should then be added back onto the empty cannon base to give the repaired cannon base.
This item cannot be taken outside of the repair area. It can be re-obtained from Isak if you have left the area after obtaining it.