https://youtu.be/hiwJgp2XuBI
PHYS 1101: Lecture Twenty-Two, Part One
Welcome to lecture 22. Let me introduce this material by drawing an analogy to a process and material we’ve already studied. Remember, we started out chapter two and chapter three doing kinematics. That was just describing motion. We now know we want to call that motion linear or translational motion, where it’s in meters. Velocity is meters per second. It’s describing the overall trajectory of an object.
We then went on from the kinematics to ask what do forces do, what causes motion to change. And remember, that ended up translating to that forces don’t cause velocity, but forces cause acceleration. They cause the velocity to change. Okay. Now we need to do that same transition, that same analog, for our rotational motion.
In other words, the previous two lectures, we’ve gone through a lot of detail with rotational kinematics, describing the angle, the angular velocity, angular acceleration of an object. We now need to take the next step and ask what causes an object to wind up, start spinning faster and faster and faster? Or wind down, start slowing down, spinning slower and slower and slower?
That’s the analog of chapter four. For rotational motion, this analog turns out to be torque plays the role of force. Torque causes angular acceleration, just like a force causes acceleration. So, all of the subtleties that were buried into this connection, you have to realize, apply, to this analog. Here’s what we’re going to end up with. We’re going to have equation that says our angular acceleration that we have, how much something is spinning faster and faster and faster, or slowing down, spinning slower and slower and slower, that’s caused by or is equal to the sum of what are called torques, divided by a new variable that plays the role of mass. This is called the moment of inertia.
Let me, under this equation, remind you, so you can see the analog. Acceleration is equal to the sum of the forces divided by m. We have the perfect analog here. So, what this lecture is about is appreciating the possibilities for the left side of the equation, the angular acceleration, and then appreciating what the right side of the equation has to be. This now becomes our new starting basic equation. This is going to deserve big highlight.
Okay. The left side. Angular acceleration, just like linear acceleration, really falls into two camps: either something is in equilibrium, or it’s not. If it’s in equilibrium, there’s no net torque that’s overall trying to spin the thing up or spin it down. So, the left side is zero, which is what I have written down here. Therefore, if the left side is zero, this denominator just cancels off, and I end up with that the sum of the torques has to be equal to zero. Okay. That’s one general category for Alpha.
If I’m in equilibrium, the sum of the torques add to zero. They have to balance. The other possibility is I’m not in equilibrium. I have a real number on the left side of this equation. Maybe for radians per second squared, minus 2 radian per second squared. Whatever it is it’s equal to, it’s caused by the net torque divided by this quantity called I. That’s what we’re going to do in this lecture, is more carefully think about what a torque is. I’ll define it for you, and we’ll look more carefully at this I, moment of inertia. To start, though, let’s do our warm-up quiz questions.
First one here, I have a fan blade that’s slowing down. It’s rotating as shown. What are the proper signs for Omega and Alpha? Do you absolutely need a net force to keep a box moving along a rough surface? Okay. Where were we?
What I have for you here is a great overview of all of the basic equations that describe rotational kinematics. We’re focusing now on an object that may have some rotational motion. I have some starting condition. The thing is rotating around. Perhaps it goes around multiple times, but then I have an end. If I want to describe the angle variables for this motion, for this object, I’m going to use angular position, angular velocity, and angular acceleration. So here I show all seven of them grouped in the way that helps me understand them. I have time, position, velocity at the start. Time, position, velocity at the end. Then, I have to have this constant angular acceleration in between.
It’s a little tricky to become comfortable with these variables. So I think the best approach is just to practice. Perhaps read a bunch of problems at the end of the chapter and just see if you can pick out the quantities and assign them to these variables properly. Memorizing the units can be helpful for recognizing the variables and what they physically represent. Position is always going to be an angle. That could be in units or pseudo units, if you will, of radians, or it could be degrees, revolutions. Angular velocity is always angle per second, per time. Angular acceleration is a unit of angle per second squared.
Once you have these variables, given their physical connection and their meaning, we know that the same three sets, or the same set of three equations for our basic kinematics, we can apply to these variables that represent just the angle quantities. So, my first one connects velocity, angular velocity at the start and the end, with the time and the angular acceleration. Equation two is the one that gives me the angle position at a later time, et cetera. Then, the third equation is the one that doesn’t depend on time. That’s sometimes useful to us. Okay.
If something’s rotating around in a circle, in addition to this angle description of what’s going on, I may want to talk about the real distance along the circular path, the real motion characteristics along the circular path. I call that… We call that the tangential variables. I have, again, an initial snapshot in time, the start of my problem. Here’s the end scope. I have a constant possible acceleration in between, and an initial and final velocity.
Again, there are my position, velocity, acceleration variables. But now, they’re linear quantities, in that it’s a position in real meters. It’s a tangential velocity in meters per second, and the acceleration is meters per second squared. With those definitions, I have my same set of three equations that I can use that uniquely connect all those variables.
The last category I have here is the connection between these two views: the angle variables and the corresponding motion along the circular path. They’re not independent. They’re related. If something’s rotating with a certain angular velocity, if it’s at a certain radius from the center, its VT is defined. It’s not unique. It has to be a certain value, et cetera. So, the connection between these two sets of variables is shown in this third panel. So, it’s connecting the rotation variables with the distance variables along the circular path. Here are the three basic connections that we need. The tangential velocity, which I show you here. It’s a real vector with direction, is r times Omega. The centripetal acceleration. That’s the part of the total acceleration vector that points to the center of the circle. Then, I have a tangential acceleration. That has to be perpendicular. That’s the perpendicular component, the part that’s tangent to the circular path.
Here’s how they connect. r times Alpha gives me my tangential acceleration. r times Omega gives me my tangential velocity. Centripetal acceleration is, just like we learned in chapter five, v2 over r. I have this note here at the very bottom just to remember if ever, whenever we have acceleration, there has to be a force in that same direction to cause it. For circular motion, it’s useful to think of the radial. It’s called the radial direction, the direction that’s toward the center.
Then the perpendicular direction to that. So I have to have a force toward the center that’s responsible for aC, that component. And I have to have a force in the tangential direction that’s responsible for at. This is the part of the acceleration that causes the speeding up, the winding up around the circle. aC is what’s responsible for maintaining a circular path for the object to follow the circular motion.
So, here’s our connection again. This is just our trigonometry, Pythagorean Theorem, that the total acceleration of the hypotenuse would just be the sum of the squares of the two sides, the two sides of this right triangle. So that one page is an excellent summary of all of our main variables for chapter eight.