https://youtu.be/vQ2c6pAeQHA
PHYS 1101: Lecture Twenty-One, Part One
Welcome to Lecture 21. This lecture is an extension of what we learned in the last lecture, very related. What we’re going to do is extend our description of some spot on a rotating object, say, that’s following this circular path. We’re going to extend our description from the angle notion that we focused on in last lecture, where we could describe the kinematics in terms of the angle position, the angular velocity, and the angular acceleration for the motion of this object, or this spot, as it rotates around.
We’re going to connect that now to the literal length motion, or what’s called linear motion, along the circular path. We’re going to be describing now exactly what we did in chapter two. This is going to be one-dimensional motion, where a position is meters along this circular path. I’m going to talk about a velocity in meters per second along the circular path, and then an acceleration. The only difference from chapter two is that we are restricted to this line that represents this circular path. Let me jot that off to the side.
So, what’s called these tangential variables for rotational motion are nothing more than the variables we need to describe motion along the circle. These are going to have units of meters, real distance along the circle, velocity, meters per second, and then acceleration along the circle, meters per second squared. This is not analogous to chapter two; it’s exactly like chapter two. We’re going to go one step further, though. We’re going to think about our position, our velocity, our acceleration along the arc, but we’re going to learn that those quantities are connected to corresponding angular quantities.
In other words, the velocity along the circular path is related to the radius of the circle, and the angular velocity. So there’ll be just a few new equations that we have to learn. I’ll highlight them for you, but in essence it’s going to be just doing one-dimensional motion along this circle. Here’s your first quiz question. It’s related to circular motion, some previous material. Pause the video for this one.
Quiz question two, again, goes back to previous material. You’re picturing a ball that’s being thrown horizontally from the top of Mount Everest. It follows…You launch it in this direction, but it’s going to follow this circular path, this circular motion. This is chapter two question. Think about the forces on that object, and think about then which statement is true. What’s the value of the ball’s acceleration?
Here’s our quick overview describing the angular motion of this object, in term of these angular quantities. We have some object rotating around, and some point on that object I want to describe the position, and how that position is changing. I’m going to do that in terms of the angle information because if it’s stuck to that rotating object, it’s going to stay stuck to following this circular path, and so I can uniquely characterize its position based on the angle that it’ll be at, and just knowing the radius of that circle.
So, last lecture was all about learning to describe this angular motion, how the angle changes with time, is it winding up, or is it winding down, etc. We had an analogy to chapter two, very strong analogy. We’re going to have a beginning scope to our problem, and an end scope. This is an initial snapshot, and a final snapshot, and we’re interested in the motion in between, or during this time interval.
At the start, I uniquely can characterize the time, position, and velocity for this angular information. At the end, time, position, and velocity. And then I’m restricting myself in this class to cases where the angular acceleration is just constant. It’s going to be uniformly winding up, or uniformly slowing down, winding down. That would be a constant value of Alpha. In other words, every second my angular velocity, from start to finish, is changing by just the same steady amount every second.
The thing to help you remember what these variables represent are the units. Instead of it being meters, meters per second, meters per second squared, it’s simple angle, angle per second, and angle per second squared. We’re going to use the unit of radian to describe those angles. So these are the SI units that we want to use for each of these quantities. I think the most difficult one to memorize, or to recognize in a problem, is the angular velocity, Omega.
When you see any unit of angle per second, angle per second is like meters per second. I think if you saw something in units of meters per second, you would recognize it as a velocity. Okay, get in the habit of doing the same thing for angular velocity. Sometimes it’s not going to look just as radians per second, but it may be, for example, revolutions per minute. That’s also angle per time. That can be converted to radians per second. So I think Omega is probably the most difficult variable to learn in this new material, and to memorize.
Once you are able to identify the three key variables at the start, in the end, and the alpha in between, you’re able to identify the variable you need to solve for. Has to be one of these seven. You then can use the same three basic kinematic equations that we’ve had from chapter two. The only difference now is that these are the angular analogs to those quantities.