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## AP®︎/College Physics 1

### Course: AP®︎/College Physics 1 > Unit 3

Lesson 4: Gravitational fields and acceleration due to gravity on different planets# Inertial Mass vs. Gravitational Mass

The inertial and gravitational mass of an object might be the same number, but they tell you two different things about an object. Learn about the two concepts each type of mass describes and how to measure each independently of each other. Created by David SantoPietro.

## Want to join the conversation?

- @3:34is the digital scale measuring mass or weight?(1 vote)
- All scales give you the WEIGHT of the objects you are measuring.

If you want to know the mass, you can just divide with gravitational acceleration according to your field.(1 vote)

## Video transcript

- [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.