Main content

## Rotational inertia and angular second law

# Rotational version of Newton's second law

AP.PHYS:

CHA‑4.D (EU)

, CHA‑4.D.1 (EK)

, CHA‑4.D.1.1 (LO)

, INT‑3.F (EU)

, INT‑3.F.2 (EK)

, INT‑3.F.2.1 (LO)

## Video transcript

- [Instructor] Alright,
so we know how to find the torque now, but who cares? What good is torque? What good is it gonna do for us? Well here's what it can do. We know from Newton's second law that the acceleration is
proportional to the force. What we would like to have is some sort of rotational analog of this formula. Something that would tell us alright, we'll get a certain amount
of angular acceleration for a certain amount of torque. And you could probably guess that this angular
acceleration's gonna have probably something with
torque on top 'cause torque is gonna cause something
to angularly accelerate. And then on the bottom, maybe
it's mass, maybe it isn't. That's what we need here. If we had this formula,
this rotational analog of Newton's second law,
then by knowing the torque we could figure out what
the angular acceleration is just like up here by knowing force, we could tell what the
regular acceleration is. So that's what I want to do in this video. I want to derive this rotational analog of Newton's second law for an object that's rotating in a
circle like this cue ball. And not just rotating in a circle. Something that's angularly accelerating. So it would be speeding up in its rotation or it'd be slowing down in its rotation. So let's do this, let's
derive this formula so that if we know the
torque we could determine the angular acceleration
just like we determine regular acceleration by knowing the force and Newton's second law. So how do we do this? In order to have an angular acceleration we're gonna need a force that's
tangential to the circle. So in order to go angularly
accelerate something you need a force that's tangential because this force is
gonna cause a torque. So let's say this is the
force causing the torque, we know how to find it now. Remember torque is R
times F times sine theta, but let's make it simple. Let's say the angle's
90 so that sine theta will end up being one
'cause sine of 90 is one. And let's make it simple too in this way, let's say this force is the net force. Let's say there's only
one force on this object, and it's this force here. Well we know that the net force has to be equal to the mass of the object times the acceleration of the object. And you're probably like, big whoop. We already knew this. What's new here? Well remember, we want to relate torque to the angular acceleration, so let's write down the torque formula. How do you find the torque from a force? Remember that the torque from a force is gonna be equal to the
force exerting that torque times R, the distance from the axis to the point where the force is applied. Now in this case, that's the entire radius 'cause we applied this force
all the way at the edge. If this force was
applied inward somewhere, it would be only that
distance from the axis to the point where the force is. But we applied it at the very edge so this would F times the entire radius. And then there's also a sine
of the angle between F and R, but the angle between F
and R is 90 degrees here, and the sine of 90 degrees is just one, so we can get rid of that. So this is simple, the torque
exerted by this force F is gonna be F times R. What do we do with this? Well look at down here, we've
already got an F down here. If you're creative you might be like, well let's just multiply
both sides by R down here. That way we'll get
torque into this formula. In other words, if I
multiply the left side by R I'll get R times F, and
now that's gonna equal R times the right-hand side. So it's gonna be R times
M times the acceleration. And this was good, look
at now we have R times F. That's just the torque. Torque is R times F, or F times R. So I've got torque equals R times M, times the acceleration,
but that's no good. Remember over here we want a formula that relates torque to
angular acceleration, not a formula that relates
torque to regular acceleration. So what could I replace
regular acceleration with in order to get angular acceleration? Maybe you remember when we talked about angular motion variables. The tangential acceleration
is always gonna equal the distance from the axis to that object that's got the tangential acceleration, multiplied by the angular
acceleration alpha. So this is the relationship between alpha and the tangential acceleration. Is this tangential acceleration? It is 'cause this was
the tangential force. So since we took the tangential force, that's gonna be proportional
to the tangential acceleration. These are both tangential here, and these forces are all tangential. That means I can rewrite
the tangential acceleration as R times alpha, and
that's what I'm gonna do. I'm gonna rewrite this
side as R times alpha 'cause R alpha is the
tangential acceleration. So this whole term right here was just tangential acceleration, and now look what we've got. We've got torque is gonna be equal to R times M, times R times alpha. I can combine the two
Rs and just write this as M times R squared times
alpha, the angular acceleration. And now we're close. If I wanted a form of Newton's second law I could leave it like
this or I could put it in this form over here
and just solve for alpha, and get the alpha. The angular acceleration of this mass is gonna equal the torque
exerted on that mass divided by this weird term, this M the mass, times R squared. And this is what we were looking for. This is what we were
looking for over here. I'm gonna write it in this box. The rotational analog of
Newton's second law for rotation is this torque divided by this term here. This M R squared, what is that? Well it's serving the same role that mass did for regular acceleration and the regular Newton's second law. And remember, this mass was proportional to the inertia of an object. It told you how hard it was to
get that object accelerating. How sluggish an object is. How resistive it is to being accelerated. That's what this term
down here's gonna be. People usually call this
the moment of inertia, but that's gotta be the
most complicated name for any physics idea I've ever heard of. I don't even know what this means. Moment of inertia. That just sounds strange. It's represented with a letter I, and it's serving the same role. It's in this denominator
just like mass is, and it's serving the same role. It's serving as the rotational inertia of the system in question. So in other words, something
with a big rotational inertia is gonna be sluggish to
angular acceleration, just like something with
a big regular inertia is sluggish to regular acceleration. So if this ball, and we
can see what it depends on. Look at, for a ball on
the end of a string, the moment of inertia for a
ball on the end of the string was just M R squared. This was the denominator. This was the term serving
as the rotational inertia for this mass on a string. And what that means is
if you had a bigger mass, or if the radius were bigger, this object would be harder
to angularly accelerate. So it would be difficult
to get this thing going and start speeding it up. But on the other hand,
if the mass were small, or the radius were small, it'd be much easier to
angularly accelerate. You could whip it around like crazy. But if the mass were very
big or the radius were big, this moment of inertia
term would get much bigger. This is the moment of inertia for a mass on the end of a string, and
that's what the I is here. So you could think about it
as the rotational inertia. That's a much better name for it. People are coming around and realizing that you should just call it this 'cause that's what it really is. This moment of inertia is
kind of a historical term. It stuck around, it's not a very good one. Rotational inertia is
much more descriptive of what this I really is. And we should note the units
of this moment of inertia, since it's mass times radius squared, the units are gonna be
kilgram meters squared. These are the units of moment of inertia, and this is the formula if
you just have a point mass. And by that I just mean a
mass where all of the mass is traveling at the
same radius in a circle. It doesn't have to be tied to a string. This could be the moon
going around the Earth. But as long as all of the
mass is at the same radius and traveling around in a circle, or at least mostly at the same radius. Let's assume this little
radius of the sphere is really small compared to
this radius of the string. If that's the case, where
basically all the mass is traveling around in a
circle at the same radius, this would be the formula to
find the moment of inertia. So how does this ever get harder? What do you have to look out for? Well we only considered one force. You could imagine maybe there's
many forces on this object. Maybe there's some other force this way. Well in that case, you just
have the net force here to make sure it's M times A, and you just have to make sure you use the net torque up here. So this formula will still work
if you have multiple torques on this object or this system. You just have to use
the net torque up here. You add up all of the
torques where torque's trying to rotate it one
way would be positive, and torque trying to rotate
it the other direction would be negative, so
you'd have to make sure signs are correct up here. And what about rotational inertia? What if your object isn't
as simple as a single mass? What do you do then? Let's look at that. Let's take this formula
here, I'm gonna copy that. Let's get rid of all of this, and let's say you had this crazy problem. You had three masses now, and one force on this mass two was 20 newtons downward, and one force was upward 50
newtons on this mass one. And they're all separated by
three meters, and can rotate. We're stepping it up, this is complicated. It can rotate in a
circle, but we can do it. We can do it with the
formula we just derived. Let's use that. This is gonna be useful. Let's say the question is
what's the angular acceleration for these masses in this
particular set up of forces? We're gonna use this formula
for Newton's second law. In angular form we'll say
that the angular acceleration if that's what we want, is
gonna equal the net torque. How do we find the net torque? Now there's two forces. Well it's not that bad. You just find the torque
from each one individually and you add 'em up. Just like you would do
with any net vector, find each individually and add 'em up. But it's not gonna be 50 minus 20. These are torques. We've gotta plug torque
in up here, not force. This has gotta be a torque, and until you multiply that
force by an R it's just a force. So don't try to just
stick this 50 up in here. It needs to get multiplied by an R. What R? Be careful, you might
think three meters, but no. The R is always from the axis of rotation which is the center all the way to where the force is applied. So the torque from this
50 is gonna be nine meters times the 50 newtons. Now we've got a torque. It's not a torque until you
multiply that force by an R. That was the torque from the 50 newtons. How about the torque from the 20 newtons? You might be like, alright I got it now. It's gonna be 20 newtons, but
I can't just put 20, right? We gotta multiply it by an R. It's gonna be 20 newtons
times, and it's not three. It's always distance from the
axis, so it's from the center all the way to where this
20 newtons was applied, and that's gonna be six meters. And sometimes when the people
get here they're just so happy they remember the R, they just do plus, and without thinking about it, but they're gonna get it wrong. You can't do that. Look at, this 50 newtons
was trying to rotate this system counterclockwise, right? The 50 newton's trying
to rotate it this way. The 20 newton is trying
to rotate it that way. They're opposing each other. These are opposite signs of torque, so I've gotta make sure
I represent that up here. I'm gonna represent this 20 newton torque as a negative torque, and that's the convention we usually pick. Counterclockwise is usually positive, and clockwise is usually negative, but no matter what convention you pick, they've gotta have different signs in here so be careful there. So that's our net torque up here. How do we find the rotational and inertia, or the moment of inertia? Well we know from the previous example the moment of inertia of a point mass that is a mass going in a circle where all of the mass is going
at that particular radius is just M R squared. But now we've got three masses so you might think this is
hard, but it's not that hard. All we have to do is say that
the total moment of inertia is gonna be the sum of all the individual moments of inertia. So we just add up all the
individual moments of inertia. In other words, this is just gonna be the moment of inertia of mass one. If that's one kilogram,
that's gonna be one kilogram times R squared. That's what this means. You take all the masses. M one, times R one squared, plus M two, times R two squared, plus M
three, times R three squared. You'd keep going if you had more. You just add them all up
and that would give you the total moment of inertia
for a system of masses. So if we do 'em one at a time, this one kilogram times the R for that one would be nine meters
'cause that's distance from the axis to the mass. That'd be nine meters squared
plus alright, mass two. If that's two kilograms, and that's gonna be times six squared. And now we keep going. We take this three kilogram
mass and we also add its contribution to the moment of inertia, or the rotational inertia, and that'd be three kilograms times it's only three meters
from the axis squared, so times three meters squared. And if we add all this up and plug all this into the calculator, we'll get that the alpha,
the angular acceleration is gonna be 1.83 radians
per second squared. So that's the rate at which this object would start accelerating
if it started from rest. It would start to speed
up in this direction and start speeding up
over and over and over if these forces maintained the torque that they were exerting. So recapping, just like
Newton's second law relates forces to acceleration, this angular version
of Newton's second law relates torques to angular acceleration. And on the bottom of this
denominator isn't the mass, it's the rotational inertia
that tells you how difficult it's going to be to angularly
accelerate an object. And you can find the moment
of inertia of a point mass as M R squared, and you could
find the moment of inertia of a collection of point
masses by adding up all the contributions
from each individual mass.