Hey everybody, let's uh we're gonna start with this last chapter here. We're starting to um up to now we've largely looked at um things that go in straight lines right. It's what we call translational or linear motion.
Um we're now going to allow things to do circles. Outside of that uniform circular motion we've done everything in lines. So this is just allowing us to investigate that circular motion a little bit more.
But we're going to remove the constraint that the angular speed is constant. So now things can speed up and slow down in circles. So we're going to remember from chapter six we talked about what we call rotation angle.
I'm going to now call it angular displacement because that's going to fit a little bit better, which we call theta. It's the angle through which motion occurs. So it's going to be the ratio of arc length.
I wrote it as s or as x. It's the length you move, the length in meters over your radius of curvature. So you can have units of radians.
Cool. The other thing we want to we talked about was angular velocity, which is the rate at things change in the angle. So it's your angular displacement.
over time, or we can also relate it to linear speed as the linear speed over the radius. The angular velocity is going to have units of radians per second. So what we looked at, you know, we've got displacement and velocity. Now we've got angular displacement and angular velocity. Angular means circular, circles.
So. So let's allow things to accelerate. So we're gonna have angular acceleration, which is a symbol alpha.
It's the rate at which the angular velocity changes. It's the exact same definition we had for regular acceleration, but now it's angular. So this is gonna have units of radians per second squared. Instead of meters per second squared, now it's radians per second squared.
Okay, seems straightforward enough. The angular acceleration we can also relate to translational acceleration, which is the acceleration we had before. The translational acceleration a is equal to the radius of the rotation times the angular acceleration.
Same format as we have for velocity. I've written it slightly different, but it's the same sort of format. Okay. The direction, since angular acceleration is a vector, the direction is going to be based on the right-hand rule, which we talked about a little bit when we did angular velocity in Chapter 6. There are two steps. You take your right hand – this is my right hand, I promise.
You're going to curl your fingers in the direction of the angular acceleration, and then your thumb gives you the vector direction. Now, this is so much easier in class because this is a teeny tiny screen. I might be backwards. We will do that in class. But here's a key piece going on here.
The angular velocity and angular acceleration are going to follow the same pattern we worked out when we had linear velocity, linear acceleration. Translational velocity, translational acceleration. Linear and translational are going to be interchanged. So.
If you're speeding up, these two vectors are going to have the same direction. Right. And then if you slow down, they have opposite directions.
Why am I why am I bringing that back up again? Great question. The direction of angular acceleration is a little bit tricky because you have to think about the way that the direction, the angular velocity is changing. It's less intuitive than velocity goes that way.
Right. So if you can figure out the velocity direction and then, you know, whether it's speeding up or slowing down, you can then quickly figure out the acceleration direction based on what the velocity is doing and speed up versus slow down. I think that's easier to this day.
So we will hit that a little bit harder in class as you have questions. So since we're talking rotation now. We're gonna...
do a lot of the same, basically all the same concepts as we've seen before, but now we're going to do them in rotation. So we can do the rotational bit faster because we have the foundation in translational motion, but now we're just doing it in circles. So a lot of the same equations show up. So this is a table that we're going to keep filling out where we have translational, rotational, and maybe an equation that relates the two of those two variables. So like you have position or displacement.
Translationally, we've called it X, Y or R, Z. Now it's a theta. And you can connect to them by taking translational equals R times theta. Right.
R is your radius of rotation. Translational velocity, you get an angular velocity. You can relate the two.
Acceleration, angular acceleration, relate the two. Now here's the coolest bit. those three kinematic equations we had before, the long, the square, and the short, we can quickly pop those into rotational versions. So the rotational version of the variables are put into these kinematic equations.
So they're, they look like wildly different equations. They're not. You just take your x and make them thetas.
You take the v, make it omega. a becomes alpha. And they're the same equations. It's a good time. So we don't have to derive those because we know them.
We know them. We're just gonna do it in rotation instead. Now you cannot mix rotational variables and linear variables in the same equation.
They don't like it. So you can go back and forth between the two using the connections. but you can't have like an x and an omega in this kinematic equation.
So we're going to practice it. And we're going to keep filling in this table as we add things. So this is one thing I want to make sure we talked about before I forgot. The rotational kinematic equations are the same set as we had for translational.
They are the same because we have, we make the same major assumption. about the angular acceleration as we did about the translational acceleration, which was that acceleration is constant. We assume acceleration is a constant value and those kinematic equations just fall right out.
Great. Easy enough. Straightforward enough at least. So let's practice it.
Right? This is pulling out like chapter two kinematic stuff. So we got a powerful motorcycle, can accelerate from zero.
to 30 meters per second in 4.2 seconds. What is the angular acceleration of its 0.32 meter radius wheels? So we've got a wheel, right? It's going to move, right? It's going to accelerate.
So this question has a mixture of translational and angular variables. Linear, rotational, how do you want to say it? So we have to be careful. What I use are the units. Meters per second, ooh, that's a V. You know, you want to use the unit to your advantage.
So let's figure out what we know, what we don't know, where we can go from there. So we have the same set of variables that we had before, XI, XF, VI, VF, A, T, but now you can do it in your rotational versions. Let's see what we have. It can accelerate from zero.
That's my initial velocity to 30. That's my final velocity. The V is because it's in meters per second, which honestly, like the initial angular velocity is also zero. It's not going anywhere. We want to know what is the angular acceleration when the wheel is 0.32 meters, right?
The outside of the wheel is doing that linear velocity. So we can use the r to relate the two. So there's two approaches here.
You can convert all these variables into angular velocities and then find the angular acceleration. Oh, we also have 10. Or you can find the linear acceleration and then convert that. Both will get you to the exact same spot.
I'm going to convert the variables this time and I'll do it a different way later. So. I want omega. Oh, I have omega i. We're going to do omega final.
It's going to be my final velocity over my radius. This equation, this is on your V equals R omega is on your equation sheet. I just rearranged it in my brain.
OK, so meters per second. Great. Meters. Great. My meters are going to cancel.
So I have 30 over 0.32. Did I do that? I did not do it that way. I'm going to calculate it.
Ready? One, two, three, two. Okay.
And I'll assign it a notation. 93.75. And it's going to be radians per second.
Where do the radians come from? Because your meters are canceling, because it's an omega, we know it has to be radians. So radians is this, like, unitless unit where you can take an in and an out as you need. Okay. So we can now find acceleration.
right, it's the change in omega over time or final minus initial over t, right. So we have 93.75 minus zero over 4.2 divided by 4.2. We get 22.3 radians per second squared. Ta-da! Great.
So it's a similar kinematic thing that we did, we've done in the past. Now it's just an angular variables. You could have also done A is change in velocity over time, change in v over t, find the A and then do the A is r alpha to find alpha. You get the exact same spot. There's slight differences in rounding.
That's totally cool. It's totally normal. I'm not worried about that. OK. I have this fishing line example.
I'm going to do this one in class. I think it'd be useful to do it together. So let's talk about this. So now that we're rotating, the distribution of your mass as you rotate matters.
So we introduce this idea of moment of inertia. It's the rotational equivalent to mass. So it's a measure of the object's tendency to maintain its rotational motion.
So if we remember... way back, inertia was proportional to mass. Like more mass had more inertia.
Now if we rotate it, we also consider, you know, where it is. So the movement, the moment of inertia depends both on the amount of mass and its distribution relative to its rotation axis. So if you spin something, although if the mass is close, it's different than if you spin it and the mass is far away from the rotation.
So the moment of inertia, I, is equal to the sum over all the pieces of the mass times its position squared. So what is this position? I'm going to define this a little bit harder.
Make sure I'm not going to write down. OK. So the distance of our mass from the axis of rotation. You might call the axis of rotation a pivot point. That would also make sense in terms of what's going on.
So you'll have a mass times its position squared. And then you take the other mass times its position squared and just add them all up. So it's going to have units. It's going to be a real wonky unit.
Mass is in kilograms. And then you got meters for your distance squared. Super weird.
I fully agree. So we're going to we're going to investigate this by doing. But we're going to add it to our table here of rotation versus translational.
So mass or rotational equivalent to mass, we have mass in the moment of inertia. And there's no real connection equation. It like the moment of inertia would be the connection equation.
Yeah. So let's practice this with point particles. The summation is assuming point particles. We'll talk about what happens when you get solid objects in a minute. So the mass, two small masses of five kilograms and seven are mounted four meters apart.
I showed on our picture here. They're on a light rod. It's a light as in is not very heavy.
We can ignore the rods mass. We want to calculate the moment of inertia. For the system, when it's rotated around the middle, so if you take a meter stick and you hold it in the middle and rotate it, part B is going to be you hold it on the end and rotate it around. So we have two masses. So we're going to have M1 and M2.
M1 I'm going to be called five kilograms. M2 is seven. Oops, that's OK.
And so the R's. are the distance to the axis from our mass. Since it's in the middle, r1 and r2 are both half the length of two meters, right? They're both two meters away from where it's rotating. Okay, so the moment of inertia is the sum over the masses times their distances.
So you have m1 r1 squared plus m2 r2 squared. Right, you just take m r squared for each piece. If we have three pieces, add another one.
If you got six, add all six of them. So we go, okay, I have five kilograms and two meters squared, seven kilograms, two meters squared. This one isn't my note. It's 48 kilogram meter square. All right.
Cool. What if we move where it rotates around? So in this, in the second case, it's now rotating off to the left.
I'm going to have you think about this one. Oh, how would I do that different? What are you going to do different?
Think that through. We're going to try it together in class. I have extra space just in case I need it. So what happens since the sum is for point particles, right?
You have discrete pieces of mass at very exact positions. What about solid objects? If I take that ruler and consider that to be the thing that I spin, what then?
What's the moment of inertia then? Well, your moment of inertia, instead of a summation, you have an integral. And then we're in calculus. So since you're not required to have calculus, you don't have to do the integration. Great.
So what we do is we provide you this table of common shapes that are rotated. So you have the moment of inertia for various shapes on this table straight out of the OpenStacks textbook. We have things like if we go to the top left, you have a hoop and it's going to be rotated around the center. So you have a hoop and it's rotated this way. You get this.
one thing. If I rotate it this way, I get something else. So the hoop about a diameter is on the bottom left.
So the axis of rotation matters, which we'll find when we do part B of the last question. So you have like a cylinder around the middle, right? It's got some thickness around the outside.
You have a disc that rotates around center here or a disc. cylinder that's gonna rotate this way. Right, it's all slightly different.
You have a thin rod. You have a thin rod in the middle, thin rod on the end. A sphere, a spherical shell where it's hollow on the inside, a slab, a rectangular slab. So all these things can be rotated.
We need the moment inertia to figure out about their rotation. We'll use those later on. We'll end up using those later on and so this table will come back. I think I have four of these on the equation sheet, four most common shapes we use. There you go.
That's what I have so far. We'll keep going next time. Make sure to do your checkpoint and I will see you in class.