Inertial Mass vs. Gravitational Mass | Circular motion and gravitation | AP Physics 1 | Khan Academy

Knowing the mass of an object actually tells you two independent things about that object. For instance, if you knew that this truck had a large mass, you'd know that it has a large amount of inertia. That is to say, it'd be very reluctant to being accelerated. It'd be difficult to speed up, and once you got it up to speed, it'd be very difficult to stop. It would take a large amount of force.

That's because it has a large amount of inertial mass. This idea of inertial mass is best exemplified with Newton's second law. So, acceleration equals the net force divided by m. This m right here, down here in the denominator, this is the inertial mass because it's telling you how reluctant that thing is to be accelerated. More inertial mass would give you less acceleration.

But mass also tells you something else. It tells you how much that object is going to interact via gravity. So, if this truck has a large mass, that also tells you its force of gravity is going to be very large. The force of gravity, Fg, on this truck is equal to mg. But this m right here is not inertial mass; this m here is telling you how much this truck interacts via gravity with other objects. That means this is the gravitational mass.

Now, in our universe, for a given object, these two values, inertial mass and gravitational mass, are going to be the same. So, this truck's inertial mass, measured in kilograms, is going to be the exact same value as this truck's gravitational mass, measured in kilograms. But it didn't have to be that way. I mean, these two ideas are conceptually different. One, the inertial mass tells you how much inertia or reluctance to acceleration something has. But the gravitational mass tells you how much that object interacts via gravity.

So, you can imagine a universe where maybe there's no force of gravity, but objects still have a reluctance to being accelerated by other forces. Or maybe you can imagine a universe where there is a force of gravity, but the number that tells you how much something interacts via gravity could have been different from the number that tells you how reluctant that object is to being accelerated. But for our universe, these two numbers are the same.

I mean, scientists to this day are still doing very delicate experiments to try to discern any small differences between these two. But, as far as they can tell, to the best experiments up to date, these two numbers are exactly the same, even though they're conceptually different.

So this is good to keep in mind. If you're going to do an experiment, you're going to be measuring either inertial mass or gravitational mass. Typically, how would you know in a given experiment if you measured one or the other? Well, I mean, if you just use a simple experiment like take a spring scale, measure the force you're exerting on a cart, and then measure the acceleration of that cart using meter sticks and stop watches or a motion sensor.

If you just plug this into Newton's second law, so if you know that acceleration from a motion detector or stop watches and rulers, and you measure the force with the spring scale and you solve for this m, well, this m is the denominator of Newton's second law. That means you just solved for inertial mass because you solved in a formula that contained inertial mass.

How would you experimentally determine the gravitational mass of this cart? Well, it's even easier. All you have to do is take a scale, you know, just a digital scale, put your cart on the digital scale, and just measure how much the scale reads. Because you know that the force of gravity is going to be measured by the scale. That's the number you get out of the scale telling you how much weight this object has.

So the scale will just read this, and if you know what planet you're on, you know what g you've got. So if you know g is 9.8 and you solve for this m, well look at you, you solved for the gravitational mass—how much this thing interacts via gravity. So whenever you put something on a scale, weigh it like that, and get m, you're getting gravitational mass. If you do the other way with Newton's second law, you're getting inertial mass.

People get this mixed up, but it's pretty easy. If you ever use a formula that involves little g or like big G, gravitational constant, big G, that means you've solved m in that formula for gravitational mass. If there isn't a g, then you're solving for inertial mass. So for instance, maybe you do some experiment where you try to very delicately measure the force of gravity between two spheres.

This would be hard; you probably wouldn't set it up like this— you’d have to be more sophisticated. But let's say you could just measure the force of gravity these two spheres exert on each other. The formula for that would be big G m1 times m of the other divided by the distance between them squared. You'd have to know one of the masses, but the spring scale could give you the force. You can measure the distance between them with a ruler. Big G, you know, it's a constant of the universe.

If you knew one of the other masses and solved for this one, you'd be getting the gravitational mass. Look at you! Use the formula that's got big G. Any formula with big G or with little g like force of gravity is mg—these are all formulas that tell you how much the object m is going to interact via gravity. Or you could even imagine gravitational field is big G m over r squared.

All of these m's here, this m here, that m there, that m there, and this m here are all gravitational mass because there’s either big G or little g involved in that fundamental equation. If there's a fundamental equation that doesn't have big G or little g, you're not talking about how something interacts via gravity; you're talking about its inertia, and that would be inertial mass.

So, for instance, if you did some other experiment—maybe you slam two carts together and use conservation of momentum to solve for m—well, momentum is mv. This formula has nothing to do with little g or big G; no gravitational constants here. So if you use this collision experiment and solve for the mass of one of the carts, you've solved for the inertial mass of the cart.

Similarly, if you use kinetic energy, this formula has nothing fundamentally to do with gravity, one-half mv squared. There's no big G or little g; this m here would be inertial mass. If you did the period of a mass on a spring is two pi root m over k, there's no little g or big g to be found in here. That means this is also inertial mass.

Unless there's a little g or big g in your fundamental equation here, your basic equation, that mass is going to be inertial mass. If there is a little g or big g, you're talking about gravitational mass. Now, if you're clever, you could do a single experiment with two phases and get both masses at once.

For instance, let's say you had a spring of known spring constant and you hung a block on it and you lowered it gently until it hangs at a certain distance. Unless you measured how much did this thing stretch, well, if you measure that with a ruler, then you know at this position the spring force kx had better be equal to the gravitational force mg.

So kx would just equal mg. If the spring constant's known and you measured x with a ruler and you know what planet you're on because g is 9.8 on Earth, if you solve for this m—look at g's right here. You multiplied by the g, and this formula came from a gravitational formula. You would have solved for gravitational mass, and now you know the gravitational mass of the object.

How could you get the inertial mass? Well, let's say you just pull down a little extra. You pull this down a little extra, you let go, and then it's going to oscillate at a certain period. Let's say you measure that period with a stopwatch. You measure how long it takes to go through one full cycle. That's got to equal 2 pi root m over k.

Now, there's no little g or big g here; this has nothing to do with gravity. So if you measure this period with a stopwatch and you know the spring constant and you solve for this m, well, now you've solved for the inertial mass of that block.

And now you know both. One stage got us the gravitational mass because it came from mg, the m did. The second stage got us the inertial mass because it comes from 2 pi root m over k, and this formula has nothing to do with gravity. So this would be a way you could find both masses at once.

So, recapping, inertial mass and gravitational mass are identical numbers but different conceptually. One tells you how reluctant an object is to being accelerated, and the other tells you how much the object will interact via gravity. If the mass shows up in a basic formula that involves little g or big g, that's going to be the gravitational mass; otherwise, it's going to be the inertial mass.