For example: a cannon, certain types of muskets etc...
Were there laws for whether women or freed slaves could not own specific weapons?
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Short answer - There were no federal legal prohibitions against weapon ownership in 1789.
Long answer. During that year the "United States" had at least 14 governments. For the first two months of the year, the territory was governed by the Articles of Confederation. I haven't reviewed all of the legal precedents under the AoC, but I'm willing to bet my collection of Stack Exchange Hats that AoC had no restrictions on gun ownership. Philosophically, that was an issue that would have been reserved for the states. For the last ten months of the year, the United States was governed by the Constitution. A quick review of the legislation passed that term, indicates that they didn't touch weapons laws. (they hadn't even passed the 2nd amendment yet). Furthermore since the clear intent of the drafters of the constitution was to avoid specifying negative rights, the absence of any mention of weapons, then there were no restrictions on weapon ownership.
But I said 14 governments, and I've only touched two. The other 12 were the state governments (Rhode Island didn't ratify the constitution till later). State laws may have restricted weapon ownership.
Second Amendent: "A well regulated Militia, being necessary to the security of a free State, the right of the people to keep and bear Arms, shall not be infringed."
In early post colonial US there was very little "Federal" law. It was mostly local (town or county) or State law. There also was no real federal "Standing Army" until the Civil War. The only military weaponry of that day were canons, and by association, large stores of Black Powder. Many local jurisdictions prohibited civilians from possessing, in their homes, canons or large stores of black powder. In Marblehead MA there was a town based "Powder House' outside the Village where the local militia had to store it's black powder, and then transport it to the fort that guarded the entrance to the harbor. The framers of of Constitution and the Bill of Rights (3 were from Marblehead MA) were also the same local leaders who enacted these local prohibitions against civilians possessing this military weaponry. It is hard to imagine Eldridge Geary or General John Glover approving of every Tom, Dick and Harry possessing the weapons of mass killing that an assault rifle with a 30 round magazine represents.