[MUSIC PLAYING]
SPEAKER 1: A lot of people had trouble with the two spools of ribbon.
So you had one of them that was tied in place, or nailed in place, I guess,
and still free to rotate.
And then the bottom one was allowed to drop.
And so you had ribbon coming out of both spools, and the bottom one
falling downwards and rotating at the same time.
And the question was about the acceleration for the bottom one.
SPEAKER 2: And so to me, the key really seems that both of them have
mass, because that makes it a step harder than just the bottom rotating.
And also, I feel like that's going to really guide the SIM.
SPEAKER 3: So let's think a little bit about the SIM, then,
before we get started.
So the system-- do we want to treat them separately or as one system?
SPEAKER 2: Separately.
SPEAKER 1: Why is that?
SPEAKER 3: Why do you say that?
SPEAKER 2: So if we take both of them as the system, then we're not going to
learn anything, because there's nothing acting on it.
SPEAKER 1: Yeah.
There's gravity pulling down on this guy, and then there's unknown normal
force here from the pin that's holding it in.
SPEAKER 3: Yeah.
SPEAKER 2: You would still have to know the interactions in the middle to
find how the center mass is moving.
SPEAKER 3: And then the next thing would be the interactions, right?
SPEAKER 1: Yep.
SPEAKER 3: So we know gravity is pulling down on the bottom spool here.
SPEAKER 2: Yeah.
And then you just have the rotation of these two, the masses.
And then the tension is going to cause both of them to rotate.
SPEAKER 3: And because the tension is uniform throughout the rope, that
means they'll have the same rotation, right?
SPEAKER 1: Yeah.
If they had different masses or different sizes, then the moment of
inertia would be different.
SPEAKER 2: There we go.
SPEAKER 1: And so then, we'd have to worry about how far out is this
tension pulling on them and all that kind of stuff.
But because everything is identical for the two of them, same tension
means same angular acceleration.
SPEAKER 2: So the interactions just match up with the models, right?
We have translation and then rotational.
SPEAKER 1: Yeah.
So the gravity will give us the net force and dynamics model.
There's your translation part.
And then the rotation part will meet the net torque
and rotational dynamics.
SPEAKER 2: So why do we actually need both of them?
Because one of the last problems we did, we found out that if something
has a force on it that, even if it makes it rotate and translate, the
translation is just described by the force you put into it.
So why doesn't gravity just drag the whole thing down--
and regular gravity acceleration.
SPEAKER 1: What do you think?
SPEAKER 3: I think it's because there's also a force
pulling it up, right?
We have the tension force pulling up, so that's going to take a little bit
away from gravity.
So that's actually a good check on our answer.
We expected it to be about the acceleration of gravity, but just a
little bit less.
So we can use that to check our answer.
SPEAKER 1: Yeah.
It doesn't matter whether gravity's pulling down on the center mass or
tension's pulling up over on the side or whatever.
It's all about adding up the total force, and it doesn't matter where
it's applied, in terms of the translation part.
We would also do have the rotation, so we do have to worry about that, too.
And that will definitely come into figuring out how to get the tension.
SPEAKER 2: We have rotation and translation, and just use those two or
three equations to find the tension, and therefore, the translation.
SPEAKER 1: I ended up needing four equations for it as a whole.
So there was for the force part, you do your force balance.
So the sum of your forces equals mass times acceleration.
I did that for the bottom one.
There's no point in doing it for the top one, because the top one's nailed
in place anyway.
SPEAKER 2: Yeah.
SPEAKER 1: And then I needed to look at the rotation for both of them.
So that was two different equations.
SPEAKER 2: OK.
SPEAKER 1: Even though they had the same angular acceleration, like we
were talking about, I still did it as two separate items--
set it up that way to start with.
SPEAKER 2: OK.
SPEAKER 1: And then the last thing was relating how far the bottom one falls
to the amount of ribbon that comes out, because the total distance that
the bottom one moves is equal to the amount of ribbon that
comes off of the spools.
SPEAKER 2: And so that's just going to involve some kinematics and geometry.
SPEAKER 1: Yeah.
A lot of the kinematics was overkill for this.
I did it through the angular accelerations.
So you have A equals R alpha as a fairly general thing.
And so when this one was accelerating downwards, the amount that it spins
tells me how much ribbon comes off of it.
And then the amount that this one spins tells me how much ribbon up
here, so the acceleration of the bottom one was angular acceleration of
the bottom times its radius plus the angular acceleration of the top times
its radius.
And that gave me the linear acceleration for the bottom.
SPEAKER 2: Due to tension?
SPEAKER 1: Due to the tension for the top one.
And yeah-- the rotation happens due to the tension on the bottom one.
SPEAKER 2: And then we can just subtract that from g, really.
SPEAKER 1: Yeah, once you find that tension, then you can subtract it off
from mg and get your total acceleration for the bottom piece.
SPEAKER 2: So when you go back and think about it, it still seems really,
really complicated.
SPEAKER 1: I know.
It's a situation where a lot of things match up nicely.
All the stuff that I was looking at this at the beginning, thinking, oh,
well, what if this one spins faster, what if that one spins faster--
it turns out they don't.