I'm working on a character for a story that has the ability to somewhat play around with gravitational forces, mainly weight reduction stuff but I'm sure there's other applications. Practically this would mainly be used for her to use a weapon that would normal be a bit too unwieldy and heavy without the power, but then I was curious because Force is mass x acceleration, which doesn't reflect weight. So would a large sword with reduced weight but the same mass still do the same amount of force when swung or potentially even more due to the increased acceleration from it being light, or am I mistaken somewhere? The only thing I can think of is that maybe air resistance or drag could affect the lighter but large weapon.
The Other Option: Messing with the Gravitational Constant
As other answers have pointed out, you could say the magic decouples gravitational and inertial masses. If you want the magic to work this way, which is likely something beyond what most consumers wish to comprehend in many genres and media, then those are the answers for you.
There is another option, though. Looking at newton's law of universal gravitation, you could say the magic messes with the universal constant to make it... Not so constant. Something like defining a local "for this set of atoms, make the constant a different value" sort of function.
This gives a weapon which is light to carry, but still takes the same amount of force to accelerate (in most cases). The benefit here is that it allows for normally fatiguing guards to be easily maintained. Things like several of these German longsword guards, especially Vom Tag, Ochs, or Langenort. (Italian and Japanese swordsmanship have parallels to a lot of these!)
This is usually not an issue for well trained individuals over "short" time periods, but then again, they are trained for it and not wielding stupidly large/heavy weapons. Additionally, these guards are usually not positions to "hang out" in, so a fighter would strike from one and move into the next guard.
The Downside of Decreased Mass
You do not want decreased (inertial) mass to make the sword easier to swing, because that means the sword is easier to deflect and block! Unless this magic dramatically increases or decreases effective mass within fractions of a second, can choose which forces from which people to experience, or creates a separate "gravitational but not inertial mass", this magic is a bad deal for weapons.
A weapon's (inertial) mass is crucial to its behavior in combat. Removing that inertial mass makes it harder to deliver required blows and the alternative, removing the gravitational mass, doesn't negate the forces required to actually accelerate the weapon.
Better Idea: Throw Things Up!
Wait, don't downvote just yet! This magic could be better used to throw projectiles, give them their weight back mid air, and see them fall with increased energy.
In the case of projectiles, you can more easily throw them as only drag would slow them down. Give them their mass back in flight and you have a sudden increase in potential energy, which means bigger impact on the other side.
Better Tactic: Use It On People
Alternatively, consider that a lot of martial arts requires a stable base. Suddenly becoming 50% lighter in combat is a big deal. It's enough to throw you off balance, trip you up, and allow for a competent foe to take you entirely out of the fight.
Additionally, becoming 50% lighter yourself can cause you to pull literally impossible maneuvers that most fighters simply have no experience with. A bounce in your step becomes a leap, your forward lunge a charge, and so on. It would take training to fully take advantage of this, but it would certainly make disengaging (a crucial part of not dying in armed fighting/duelling) really easy!
For the purposes of your character, and since you've included the "magic" tag, this can work any way you want it to to fit your story. Perhaps your character can decouple gravitational mass from inertial mass, or perhaps they can locally flatten space-time so particular objects don't feel (as much) gravity, or perhaps they only think of it as a gravity effect but in fact they have a (psychologically?) limited form of telekinesis that lets them handle some heavy objects as if they were much lighter.
In the last case, they might be able to pick up a two-handed greatsword as if it were a small wood dowel, swing it the same way, but still have it impact with its full weight, potentially (given skill, senses, and reflexes to back it up) making them a whirlwind in melee combat.
Using this as a weapon is pretty obvious. Lift the hammer with weight set low. Bring it down with weight set high.
So your hero can lift, say, 30 pounds without the special effect. But 1000 pounds with the effect. Just to have numbers to think about. And also to have a size to think about, 1000 pounds of steel is round about 2 cubic feet. That is, a not very large aquarium sized object. Roughly 60 liters for the metric fans. About 16 US gallons.
So they lift the huge weapon. Then just turn off the "juice" when the object is over the enemy's head. Smack! Make sure to step back and not let your own weapon hit you.
If they get in a good hit, it's time for mead and cakes with the other supers. If they miss, well. There's now a 1000 pound hunk of steel on the floor. The enemy isn't likely to be picking it up and hitting back. Meantime they can choose alternative weapons. Like small cars, furniture, other superheroes, etc.
Depending on how much it "takes out" of the hero, this can have lots of other possible uses. It's a Dungeons and Dragons trope to be able to use this sort of thing as a trap. Imagine being able to lift a ton of any cheap material. Sand, for example. And carry it up some stairs, and place it behind some barrier. And arrange the barrier to be spring loaded and open when the victim steps on the trigger. Suddenly they get a ton of sand dumped on their head. If one ton is not enough, then make twenty trips. If it does not tire the hero, it's just walking up a flight of stairs.
Making a waterfalls run backwards can have lots of interesting effects. This town over here gets on your nerves? Flood them out.
Let me present two situations for you.
Pick up a large book, and put your other hand on top of it. Shake it up and down. The difference between the "up" and "down" is gravity.
Take the same book and hold it vertically between your hands. Shake it left and right. What you're experiencing there is momentum. That feature will exist regardless of the gravitational field it's in.
If your power just effects gravity, a weapon that has no weight will still have momentum.
- It will take less effort to lift
- It will require most of the effort to swing.
- All impacts, whether they're from above, side, or below, will have the same force as a sideways swing.
- Overhead swings will not have weight advantage, and that's most of what you get with an overhead axe or war hammer swing
- Underhand swings will have more force than expected, but you still have to worry about hitting the ground
- It will provide no advantage with accuracy, as that's a momentum and reaction-speed thing.
Given all that, you'll be able to use polearms to greater "poking" advantage. Opponents would be expecting to be able to knock them down, but will be faced with the weapon's full momentum with no gravity assist.
In strictly scientific terms, the gravitational mass is the mass that we measure with gravitational effects, that is how strongly a given mass attracts another mass, while the inertial mass is the quantity we measure with the formula you quote, that is how much a mass resists the effect of a force. They are the same because of relativity, but with different physics they might be independent.
Therefore if you want to be strictly scientific, your hero when swinging around an oil pipe with this power will have the same effect of swinging a pool noodle.
However, if your character can decouple the gravitational and the inertial mass, she changes only the gravitational effects but not the inertial ones and can very plausibly wield a massive weapon without worrying about its weight and not giving up the inertial effects associated with it.
However the above messes up with relativity, so stay away from it in your story.