Inertial Mass vs. Gravitational Mass

- [Instructor] 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 'cause it has a large amount of inertial mass. And 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 being accelerated more. Inertial mass would give you less acceleration, but mass also tells you something else. It tells you how much that object is gonna 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. So the force of gravity FG on this truck is equal to MG that this M right here is not inertial mass. This M here, is telling you how much this truck interacts via gravity with other objects. And that means this is the gravitational mass. Now in our universe for a given object, these two values inertial mass and gravitational mass are gonna be the same. So this trucks, inertial mass measured in kilograms is gonna be the exact same value as this trucks, 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 could imagine a universe or maybe there's no force 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 decern any small differences between these two. But as far I 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 gonna do an experiment, you're gonna 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 stopwatches or a motion sensor. And if you just plug this into Newton's second law, so if you know that acceleration from a motion detector, stopwatches and rulers, and you measure the force with the spring scale and you solve for this M well, this is the denominator of Newton's second law. That means you just solve for inertial mass, 'cause 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 take a scale, you know, just a digital scale, take your cart, put your cart on the digital scale and just measure how much the scale reads because you know that the force of gravity is gonna be measured by the scale. That's the number you get out of the scales telling you how much weight this object has. So the scale would 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 solve 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, (mumbles) 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, M one 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 you could 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 gonna interact via gravity. Or you could even imagine gravitational field is big GM over R squared. All of these M's here, this M here, that M there, that M there, and this M here all gravitational mass, 'cause 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 by gravity, you're talking about it's inertia, and that would be a natural 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. So unless there's a little G or big G in your fundamental equation here, your basic equation, that mass is gonna 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 got 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 it this position, the spring force KX had better be equal to the gravitational force, MG. And so KX would just equal MG if the spring constance known and you measured X with a ruler, and you know what planning you're on 'cause G is 9.8 on earth. If you solve for this and look at G is right here, you multiplied by the G and this formulate came from a gravitational formula, you would have sold 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 gonna 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 two 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, 'cause it came from MG the M did. The second stage, got us the inertial mass 'cause it comes from two 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. And if the mass shows up in a basic formula that involves little G or big G that's gonna be the gravitational mass. Otherwise it's gonna be the inertial mass.