https://youtu.be/dyWEUB4_r5E
PHYS 1101: Lecture Twenty, Part Two
Now we’re ready to move on the Chapter 8. As I say, it’s the “Kinematics of Rotating Objects.” Let me show you the tree picture here to give you an idea of where this fits in.
We just are coming out of Chapter 7, and we learned to use two main equations to solve two types of problems involving the ideas of momentum. There’s the impulse momentum theory, and then there’s the conservation of linear momentum. In this class, we’ll just call it “conservation of momentum.”
This was a whole new set of problems. We’re going to be working really in a new area here of this kinematic section. Instead of looking at the kinematics along a line, or even in two dimensions where it’s a curved trajectory, we’re going to be doing it just a specific subset here of rotating objects. It’s just a special case.
We’re going to learn that we can still mathematically describe the path that part of this object follows, and the length of that path, and the speed along that path, etc. But Chapter 8 is about a whole different way of characterizing that motion. We’re going to focus on the angle position of this object.
If it’s fixed and rotating at an axis, and it’s rotating about that axis, we’re going to look at, say, a spot on a wheel and we’re going to be able to talk about the angle position that spot is at, and the angular velocity, and the angular acceleration.
Here’s the equation picture. For our kinematic section, we had Chapter 2, which was for linear motion, that’s motion along a line, and we had three fundamental equations. We had to memorize what these variables specifically represented, and then be able to identify them in a problem. Then we could solve a whole range of motion-along-the-line problems.
We then moved on to Chapter 3, where we had the same three sets of equations for the horizontal parts of our vectors, and then the same three sets of equations for the vertical parts to our vectors. We worked problems where we could have a curved trajectory.
What we’re going to do now is very analogous to Chapter 2, linear motion. We don’t have to worry about components of vectors. It’s just going to be like one-dimensional motion, Chapter 2. It’s going to just involve variables that describe angular velocity, angular position, angular acceleration.
Let me show you what those three equations are. Here I’ve copied them over. Rotational motion now. Object or a spot on a rotating object, we’re following the motion of it. Like here, we were following the motion of that object along a line.
The three same equations are going to work for us, but we need a whole new set now of seven variables. I’ll summarize it again for you later. The analog of time, position, and velocity at an initial instant, a time, position, and velocity at a final instant, and then an acceleration that is constant between those two times. Now, it’s angle position, its angular velocity, it’s angular acceleration.
Our first equation is going to just tell us how the angular velocity changes, because something is winding up or winding down. The second equation tells us how the angle position depends on my initial value’s time. The third equation, again, is the one that doesn’t involve time, but connects how the angular velocity relates to what it was initially, and angular acceleration, and angle positions.
This is worth highlighting. Let me just put a yellow box around it. These are going to be our starting equations. To be able to use them, we need to be able to sketch a picture, and we need pull out from the problem the knowns, which of these variables we have values for, and what variable we have to solve or focus on in order to determine the quantity that we want.
It’s just another subset of this red section, another collection of leaves here, a different collection of problems.
Here it is said a different way, to give you a different picture. Here are our seven variables. Let’s look at them first in the context of one-dimensional motion along a line. At the start of a problem, we had time, position, velocity.
I’m going to color code them here to try to help solidify it in your mind. Position, let’s use blue. Velocity is always green. Time we’ll keep as black. We had time, position, velocity at the start.
At the end scope of our problem, we had time, position, and velocity. Then, for the duration, the time interval completely between, I had to have a constant acceleration. It could be 0, or it could have been a constant number.
Then here are my three equations that connected those seven variables. The name of the game was when I read a problem, sketch the scene, define an origin, do whatever it takes so you can decide what these variables are for that particular problem, and what variable do you have to solve for. Then, use these equations, leverage them, to do the algebra and solve it.
We’re going to do the same thing over here. It really is one-dimensional motion. We just have these three equations to work with. I’ve got a whole new set of variables, and you’ve got to become comfortable with them and memorize them.
Here’s the picture. Now it’s going to be an object going around in a circle. I’m, again, going to have a start to my motion I’m describing, an initial snapshot in time. I’m going to set that time to 0.
Then the clock’s going to tick away, and then there is going to be a final time, or an end to the motion I’m trying to describe. At each of these instances, I’m going to have variables that capture time, position, velocity. At the end, time, position, velocity.
Now, it’s angle, angular velocity. My analog to acceleration is the variable alpha. It captures the angular acceleration, so it has the constant value from the start of the problem until the end. This captures the essence of winding up or down.
If something is rotating at a steady rate, it would not be winding up or down. It would have an alpha of 0. Only if it literally is starting to wind faster, and faster, and faster, and faster. Or it’s winding down. It’s going slower, and slower, and slower, and comes to a stop. Only then would you have an angular acceleration.
I’m going to say a lot more about those variables and what you have to substitute in for them, what the units are, here in a minute. Let me contrast Chapter 8 for you compared to Chapter 5, where we also did rotational motion.
Chapter 5 was called “Uniform Circular Motion.” If you go back to that chapter, you realize that what we were doing was focusing on, say, an object that’s following a circular path. The speed had to be constant. It could not be winding up or winding down. It had to just be going around at a nice, steady pace in a circle.
Now we’re going to do a little bit more involved circular motion. Now the object can be a solid object that’s rotating about some axis. Right here for you, this is a rotation axis. This could be like the wheel of your car, a record player, if any of you know what that is. Something that has a clear point that doesn’t rotate, so it’s got the center of this rotation motion. Then it can be any-shaped object that’s rolling around that point.
We’re still going to be focused on, and have to identify, a specific spot that we’re interested in describing the motion of. My object, which I tend to try to always sketch with a red pen, is still going to be red. It’s going to be a spot on this rotating object.
I’ll always indicate with an arrow in some way the direction of rotation. We’re going to be describing the angular motion of the spot. Here are the variables we’re going to use to do it.
Let me first get my blue highlighter. As we did with position in describing linear motion, we need an origin. All these problems, it’s up to you to define where you want that 0 to be.
I like the convention of let’s just draw a sketch of this rotating object, start at the center, and draw a horizontal line straight to the right. We’re going to call that 0. If our object is at an angle from that 0, then it’s the angle theta. That describes the angular position.
For the notion of position, it’s the angle that we care about. We call that the Greek symbol theta. We’re most familiar with degrees being called an angle we’d use to describe that. If this red spot we’re interested in were sitting up here, the angle position would be 90 degrees.
Now, the next level of complication is to imagine that this spot is moving around at a steady pace, undergoing uniform circular motion. That question is asking how fast is it going around? Let me make that green to match the notion of velocity.
It’s the angle velocity we’re going to be using. It’s the angle per second that gets swept out. For example, 10 degrees per second. Every second, the angle advances by 10 degrees. This is the symbol, omega, we use to describe that. It’s a Greek symbol. It’s not “W.” It’s omega.
The next level of complication is this acceleration analog. Is this a record player where you’re just turned it on? This scratch on the record started out at rest, but now as the motor starts winding up, this starts speeding up, as it gets up to its playing speed. That would be an example of winding up.
Let me highlight that in red because red is our color we use for acceleration, of course. Now, it’s the angular acceleration, and alpha is the variable. It’s a Greek symbol. That is what it looks like, alpha.
Alpha tells us, every second, how omega changes. Omega tells us how theta is changing.
Let’s walk through this analogy to solidify in our minds what these variables represent. Let me show you what we’re going to do in the context of linear motion. Then I’m going to ask you to do the exercise for angular motion.
We’re going to be comparing here something moving along with constant velocity. The rotational analog is uniform, circular motion.
Let’s start on the left. My motion diagram for constant velocity would be equal-spaced points. This object’s moving to the right. It has a velocity of +2 meters per second, and it looks like I have it starting here at the origin, so its initial position is 0.
What I have in this little table that we’re going to fill out is for these values of time we’re going to fill in what the X coordinate is, and then what the velocity is at each of these times.
Well, the velocity one I’m going to do first because that’s easy. If this is my motion diagram, every second the position is increasing in equal amount. By how much? Two meters per second. My velocity doesn’t change.
for all of these time intervals, my velocity is constant, as it says here, “constant velocity.” Two meters per second is what it is the whole time.
What’s my X coordinate? Well, if I started out at 0, one second later, I have advanced two meters. One second, I’m at position two meters. Another second goes by, and I advance another two meters. Now I’m at four meters. Another second goes by, and now I’m at six meters.
One way to see that another way is to take the second kinematic equation, which says X is equal X(0) plus V(0) times T plus 1/2 AT-squared. I have constant velocity, so this is 0. This is two meters per second. My initial position was 0.
When you evaluate, X is equal to two meters per second times time. You’ll see for T equals 1 second, 1 times 2 is 2. T equals to 2 seconds, 2 times 2 is 4. Then 3 second, I get 6 meters. You’re actually using this equation.
You also can just think it through. Every second that goes by, I’ve got to advance by two meters.
You’re going to do that exercise for uniform circular motion. My angular acceleration is 0. My motion diagram shows that, for every second, the angle of this object is advancing by the same amount. Notice all these wedges of pie are the same size slice.
I’m told that my angular velocity is 30 degrees every second. At T equals 0, when the clock starts, I have shown in this sketch that the object is sitting in an angular position of 0 degrees.
One second later, it has advanced, and this angle now, after one second, would be 30 degrees. After another second, my angular position is this. After another second, my angular position is that.
You’re going to have text boxes to enter your answers into in WebAssign. For Quiz Questions 5, 6, and 7, work out what the angle position is. For Questions 8, 9, 10, and 11, I want you to fill in what the angular velocity is.
Now we’re going up to the next level of complexity. I’m going to again start out on the left, with the linear motion, and walk you through that analogy. Then you’re going to fill in the numbers for something that’s undergoing the same kind of circular motion.
The next level of complexity is now where I have some acceleration. We’re just restricting ourselves to 1D motion here, so these are all in the horizontal direction.
Simple case object starts out at 0, when the clock starts, and I’m told its initial velocity is 0. It starts speeding up. Its acceleration is two meters per second squared. That tells me, as I like writing it this way because I can physically picture what it means, that every second that goes by, per second, my velocity changes by +2 meters per second.
Let’s fill in some values here. What’s my initial velocity? Well, that’s 0. I was told that. Initial velocity is 0. What’s my acceleration? Well, here it is. It’s a constant value the whole problem long of two meters per second, every second.
I’m going to go down into this whole column, note that it’s +2 meters per second, every second. This is how much my velocity has to change every second.
Let’s use that information, and I can right away go up here and map what my velocity has to be. It started out 0, but every second, it has to change by +2.
One second later, I’m at +2 meters per second. The next second, I’ve got to add two to that. I’m at four meters per second. Then I’m at six meters per second.
What about my position? What values do I fill in for this? Well, let’s just look at the motion diagram first to appreciate something. Let’s consider that this is a second between all these time intervals. Notice that the object is getting further and further every second. That’s because I’m speeding up.
These numbers aren’t just a simple multiplication as we pointed out before. Let’s go to our kinematic equation and see if we can work out what they are, just like we did above.
This equation will tell us what these position values are when T equals these different values. My object started out at rest at the origin, so these two terms are 0. The equation becomes very simple. I have 1/2 times AT-squared.
Let me plug in numbers for that. One half times my acceleration, two meters per second squared. Then I have to multiply by time, but then I have to square it.
Let me point out, just simplified a bit there, that that two happens to cancel with my value of two for acceleration. It looks like the real value for X in meters is simply going to be the number of seconds squared.
At T equals 1, 1 squared times 1 is 1 meter. When T is equal to 2, for this category, 2 squared is 4, times 1 and I’m up to 4 meters. At T equals 3 seconds, 3 squared gets me 9, times 1 and I’m at 9 meters.
You can see this progression. When I’m speeding up here, I go from the origin to 1 meter, then 4 meters, then 9 meters. Here’s 1 meter, 4 meters, 9 meters.
My position is increasing “quadratically” as people say. It’s quadratic in the time. The time variable is squared.
You’re going to do the analog of this now for the angular motion. Let’s be sure you see first that it’s exactly the same thing you’re comparing.
Here’s my origin. The object starts there. I’ve got an angular position initially that’s 0. Now the thing is winding up. Every second, its angle gets bigger, and bigger, and bigger. This would represent just turning on your record player. It has to start from rest, and it’s got to increase, wind up, to get up to its running speed.
My initial angular velocity is 0. This is like a record player starting at rest. That captures that effect, omega 0, 0. My angular acceleration now says that omega changes by 20 degrees per second, every second.
I’ve already written in here for you that alpha is 20 degrees per second, every second. This tells me if omega started out at 0, one second later, it’s got to be 20 degrees per second. That’s its angular speed at that instant.
Two second later, I’ve increased it by another 20 degrees per second. Fill in this column first. Then follow this analogy, as I did on the left, to work out what the angular position is at each of these times.
As I said before, these aren’t equal slices of pie anymore. The slices of pie are getting bigger and bigger every second, because it’s winding up.
Next important topic. We’ve got to think about the analog of direction. Remember when we were working with vectors, we had to be very careful and indicate the direction of those vectors in addition to the value.
By analogy, when something is rotating, we want to be able to distinguish whether it’s rotating clockwise or counterclockwise. It is vector-like information then that we want to be able to handle.
Before, to capture the direction, we just used an arrow on the page, and then we defined a convention that if an arrow pointed to the right, it’s going to be positive, for example, or left, negative, etc. We’re going to follow that same analogy.
Let’s start with the arrow idea. I’m going to sketch, whenever I’m given a problem, something to indicate this rotating object, and the angular velocity. That’s the analog of the linear velocity. That arrow literally has to represent the direction the thing is rotating, at that instant.
If you glance at it at that instant, and it’s rotating counterclockwise, your arrow for omega has to curve counterclockwise. If the thing is winding up, omega is going to be getting larger and larger, alpha then has to point in the same direction.
The case shown here, alpha’s pointing in the opposite direction. That’s the analog of it winding down. That’s summarized here by these bullets.
First bullet, the omega arrow always is in the direction of rotation at that instant. Sketch yourself then an alpha arrow if the thing is winding up or winding down. This means it’s starting to spin faster, and faster, and faster, or it’s spinning slower, and slower, and slower.
You remember, I hope, that if something was speeding up along the line in linear motion, that the V and the A arrows had to point in the same direction. If something was slowing down, I had to make those vectors point opposite. That was the only way the mathematics would work out, because then the signs would be opposite.
Same analog here. Something’s winding up, sketch on your graph. Alpha has to be in the same direction as omega. If something is winding down, alpha has to be opposite to omega.
Notice, warning, warning, notice. At this point, I’ve said nothing about the sign of omega or the sign of alpha. All I’ve talked about is the decision about the direction that these arrows have to point. We’re going to get to the sign tool that we’re going to apply to this next.
Just read the problem first and be sure your arrows follow this convention. Regardless of the sign convention at all, omega has to, for sure, point in the direction it’s rotating at that instant. Fact.
Next fact, if it’s winding up, you better make alpha point in the same direction. If it’s winding down, make alpha point in the opposite direction.
Now enter the mathematical tool of a sign to capture the direction of these vectors. That’s the second red bullet here. This is the convention that’s always used for circular motion.
Picture your rotating object. Here’s your axis. An arrow that points in the counterclockwise direction, as I show, those arrows are positive. It’s the direction of the arrows again. It’s not the size of a quantity. Again, direction of the arrow for the quantity.
Sign of omega, it means the direction of the omega arrow. Sign of alpha, direction of the alpha arrow.
Let me point out some consequences then for alpha. Important. Note.
If something is winding up, alpha and omega have to have the same sign. Is it winding down? If it is, alpha and omega have the opposite sign. There, I just tidied that up for you.
If it’s winding up, alpha and omega, same sign. If it’s winding down, they have to have opposite mathematical signs. That’s the only way that those kinematic equations are going to capture the motion that’s really happening.
Again, it’s counter-intuitive. The sign, the mathematical plus or minus, indicates the direction of these arrows. Not how large it is, or not directly if it’s winding up or winding down. They just have to have opposite sign.
If something’s winding down, alpha’s not necessarily negative. Or winding up, not necessarily positive. Winding down, not necessarily negative.
Quiz Question 19. I show you an object and four dots of a motion diagram, and ask you, “What’s the sign of omega for this object?”
Question 20. Look at this picture that’s shown of this motion. Again, another angular motion diagram. First, is this object winding up or winding down?
Then, Question 21 is, “What’s the proper sign that you need to apply to the variable alpha to mathematically represent this behavior?”
We’re so close to being able to work through some examples now. Let me do the next definition for you.
I introduced to you these variables of angle using degrees, because that’s what we’re most comfortable with or familiar with. Then I think it’s easier for you to build some intuition of what these variables represent. Science, it’s more convenient to work with a unit that we call a “radian.” From now on, angles are going to be measured in radians.
If you’re given an angle in a problem at so many degrees, you may have to convert it. You will have to convert it to radians. Or maybe they want an answer in degrees, in which case the math may give you your initial answer in radians, and then you have to convert it, perhaps, into degrees.
Here’s the conversion factor. There are 2pi, 2 time 3.14, radians in one revolution, in 360 degrees. Let me highlight in our different color code here.
For angular position, instead of degrees, we’re going to work with the unit of radians. I often use this shorthand “rad” for that. For angular velocity, instead of degrees every second, we’re going to use the units of radians per second. Alpha, instead of it being degrees per second squared, we need to convert it to radians per second squared.
Here’s where it comes from, or here’s the definition. An angle in radians, it really mathematically just means the ratio of an arc length, in meters, divided by the radius of the circle of the circular path. Let me highlight this for you.
It’s that length there. That’s S, the arc length, divided by the radius, R. That would be the distance from this rotation axis out to this circular path that this object’s carving out or following.
Notice that the distance along this arc is meters. This is a real length in meters. Technically, an angle in radians really has no units. It’s meters over meters, and that cancels out.
I, however, always just write down radians, just so I can kind of do bookkeeping with my units and be sure units are consistent. Technically, radians isn’t a real unit like meters, seconds, or kilograms or something.
Let me point out, or show you, why 2pi radians is the equivalent to 360 degrees, or one revolution. Let’s think about what the arc length would be if we did go around a full circle. Let me blow this up here, real quick, for you. I want to show you, but it’s not that clear here.
This angle is our theta. That’s the angle theta, and that angle is the length of this arc divided by the radius. What if we did want to represent 360 degrees, going around one complete time? That would say that our angle, 360 degrees, or one revolution, is equal to the arc length divided by the radius.
What is the arc length? If I go around one time, that’s the circumference of a circle. That’s 2piR. Let me do it here. We’re plugging in for the angle. If we go around one time, the angle will be one revolution. That’s 360 degrees. That, in radians, is S over R, and going around once, S is 2piR. Then I still have to divide by R.
You see that the Rs cancel, so in radians, one revolution is equal to 2 times pi. These are important conversion factors that you need to remember. You’ll get comfortable with them quick.
Sometimes you’re going to be asked about the total number of revolutions an object goes through. That’s an angle it’s asking you for, but it’s wanting it in units of revolutions. Your equations that you work with are going to be in units of radians.
This is the analog of converting everything to our SI units, but then you have to use these conversion factors to get your answer into revolutions. You’ll see that in a minute how that works.
Here we are. We’re going to start doing a problem now. My next collection of quiz questions is going to be about identifying variables, and using our kinematic equations.
For Question 22, you’re going to enter the value in the answer to this problem. I would wait to do that until we’ve done the remaining quiz questions here.
At the top, I summarized the variables for us. Here’s the name of the game. Here are our three kinematic equations that describe the angular motion of this problem. I have to translate my problem into these variables. I have to work with these equations to solve for the variable I need and then translate that back into real life.
Here’s the list of variables. One, two, three, four, five, six, seven, if you count T(0), the initial clock. Let me write that down here. T(0) equals 0, and then there’s a final time.
I have, at the start of a problem, time, position, velocity. At the start, time, position, and velocity. At the end of the problem, I have that snapshot of time, that value of T, that angle position, and the final angle velocity. Then I have alpha, the angular acceleration between start and stop.
Those are the seven variables I have to think about. For angular motion, we’re going to get two of them for free set to 0. Let’s always start our clock at 0 and let’s always start our initial angle at 0.
Question 22, here’s what it reads. “I have an electric generator turbine that spins at 3,600 rpm.” RPM stands for revolutions per minute. Note. This is units of angle divided by time. Angle over time.
If you think about the units of these quantities, think about the units of your variables, it often helps you. You may have to do some conversion, because I know I’ve got to get omegas, for example, into radians per second. Note that omegas are angle per time.
There, I’ve just filled that in for you a little bit more so it’s helpful to you. Angular velocity always has to have units of angle per time. Angle has units of radians or degrees. Alpha, angular acceleration, has to have units of an angle unit per time squared, like degrees per second squared or radians per second squared.