https://youtu.be/GtwCbP5KhFs
PHYS 1101: Lecture Fourteen, Part Three
Let’s start the new material by just going over some very basic concepts about it. I’m going to be reiterating what I said at the beginning of the lecture in more detail.
The title of this chapter is “Uniform Circular Motion.” Okay. This is not studying anything new. We’re applying the same tools, just in a different context.
So there’s only a few new things we need to appreciate in this new context. First of all, the analysis that we’re going to do is exactly the same.
We’re applying Newton’s Second Law. We’re looking at the acceleration for an object that’s undergoing uniform circular motion. We’re going to be looking at all the forces on that object. That divided by m gives us the magnitude of a. And the direction of a has to come from the direction of this vector sum.
Remember this numerator, after you do the vector sum, we call that the “net force.” The direction of the net force has to be the same as the direction of a.
If we have an object that’s moving in a circle, rounding a corner, any curved trajectory, we know that the velocity is changing. If you have a changing velocity — I’ve got three examples here. The velocity in this case gets larger and the object turns to the right.
This case, it looks like the speed stays the same, but the object is headed down. It then turns and now is headed down and to the left.
This object also looks like its speed approximately stays the same. It was headed up and to the right. It turns and is now headed down and to the right. Both of these cases, the velocity is changing, either simply the direction or in the first case here, both the direction and the magnitude.
When you go through that analysis to visualize the vector delta-v, you arrive at the conclusion that you have your delta-v or your a looks approximately like this for the first case. In the second case, I’m going to dash my first vector again like I always do at the next point. Didn’t draw that very well. And then this has to be — therefore this has to be my delta-v or I can think of it as my acceleration vector.
Let me do the last one. If I dash this same vector again at the next point, so this is the velocity that it would be if I didn’t have acceleration, constant velocity would give me this. I know that’s not what happens. My velocity changes. This arrow from that tail to that tip is the delta-v or my acceleration.
Okay. Any time I’ve got a curved trajectory then, I have to have a delta-v. Therefore I have to have an a. What are all of the logical consequences of that?
The first thing is, of course, I have acceleration. Therefore the object is not in equilibrium. Acceleration vector is always going to point to the inside of the corner. You appreciate this the more you practice this visualization of the motion diagram and this vector determination of the direction of a.
If I round a corner at constant speed, this vector exercise that you do will always end up with an acceleration that points straight to the center of that corner. That’s the case here. And you saw that’s the case here.
If I have speeding up in addition to changing direction, I end up with an acceleration that points mostly toward the center, but there’s going to have to be a tilt and a stretch of that vector to represent the speeding up part. That’s going to end up tilting this vector in the direction of the larger velocity, representing the speeding up, for example, here.
But this general point is always true. It’s going to be toward the inside of the corner. Constant speed as I round the corner. It points directly in fact toward the dead center of that corner.
Okay. If we have acceleration then, Newton’s Second Law is where we want to start to analyze this motion, because this defines the constraint of what that a has to be and the result is because of a net force on the object.
So let me emphasize. If I’m rounding a corner, I’m never in equilibrium. So the left side is never 0. Okay? The only time the left side is 0 is if the object is stationary or it’s moving along in a straight line.
The next logical conclusion I can come to then is that the right side is never 0. If I have a net or I have an acceleration for that object, I have to have a net force. Direction of a comes from the direction of Fnet. If a pointed toward the center, I need an Fnet that points to the center. The value of a — it’s not 9.8 meters per second squared. That would be a special case, a strange, unique coincidence. In general, the magnitude of a has to be set by the magnitude of the net force divided by the mass of that object.
Here’s some important points to emphasize, to clear up common misconceptions.
First point. Very important. All of the exercises that we’ve done, and we’ve been through, to identify forces that are on an object, all of those still apply. We are not introducing any new concepts about new forces.
This means an object in a curved trajectory, an object sitting there, any object, what are the forces on it? If it’s on the surface of the planet Earth, I have the long range force of gravity. Every other force then is because something is in direct contact with that object.
So go around the surface of that object. Every place that something touches the object, there’s a force there. Consider the nature of that contact. Therefore the direction of that force and the value possibly.
What we’re going to see with this circular motion, objects going around a corner. For example, is it usually there’s only one force on the object that does end up pointing to the center of this trajectory, of this corner.
Therefore, that has to be the force that’s responsible for the acceleration. Every force has to be either gravity or from a real agent that’s in contact with the object, so this force that’s causing our acceleration, you will see will always be caused by some real agent. A rope. A surface normal force. Friction. A real, identifiable force that we’re used to working with.
Okay. Let me go on then with some common misconceptions, which rightly so, I think, are a result of poor wording on the textbooks’ part. In fact, every physics textbook I’ve ever read describes this in a misleading way.
Books, and in physics, scientists like to use the term “centripetal acceleration.” The “centripetal” there just means “center directed acceleration.” It says what I’ve been pointing out so far in this lecture.
The way they introduce that topic seems to suggest that it’s some new special kind of acceleration, but it’s not. It’s the same acceleration we’ve been seeing. It’s the delta-v. It’s the change in velocity that we’re seeing in the motion diagram for that object.
Okay? So nothing new about it. However, the books tend to take the variable a, which is what we’ve been working with, and they like to now use the variable a sub c, further coaxing you into thinking this is some new acceleration. It’s not new. Nothing special about this a c.
This quote, centripetal acceleration is nothing more than what’s on the left side of Newton’s Law. That a is nothing more than this vector difference between two velocity vectors. It’s this a is the same as the left a.
The next common misconception problem with textbooks, if you will, in my opinion is that they go on to use the term “centripetal force,” again suggesting there’s some new type of force that you need to be aware of or consider. There is no new force.
What they mean by “centripetal force” is nothing more than the Fnet that we have to have on the right side of Newton’s Second Law. The net force that we know has to be there to cause this acceleration that points to the inside of the corner. Every force, Fnet included, we know always comes from a combination perhaps of forces that add up to an Fnet pointing to the center of the corner, or what we’ll see often is that there’s only one force that’s in the direction that points to the center.
The other forces on the object all balance out. So Fnet effectively is caused by only one force. And if you’d like to, the book would like to then in addition to its label as perhaps a tension force or a friction force, you may also refer to it as a centripetal force if you like, but there’s nothing new.
Okay. Let me quickly walk you through what I mean by that using a simple example.
Let’s look at this and analyze this situation. Imagine that you’re swinging a ball around on a rope, but it’s sitting on a table. So at this snapshot, the ball happens to be passing this location. Let’s think through Newton’s Second Law analysis. What’s the acceleration of that ball at that instant? And what forces are on that ball?
Let’s draw a free body diagram. When you’re doing circular motion, you will always have to decide what instant it is you need to focus, because as time goes on, we know that the direction of a changes, right? Because it always has to point to the center of the circle.
Therefore the direction of forces are probably changing. Need to decide what instant you’re supposed to focus on. It’s either defined in the problem or it will be defined in the problem like I’m telling you here. Look at this instant.
Further then, at that instant, when you go to identify and put all the forces on the object, it’s usually helpful to use the side view perspective, because picturing that you’re standing beside the table and you’re watching this ball swing around roughly horizontally, you’ll have a good, from that perspective, a good ability to draw in the forces that you need to consider.
So side view. Let me jot that down.
So here’s my dot for the object. Let’s draw the forces. At that instant, the ball has weight. Force due to gravity pulling it down. I know that the — as I then go around the object and ask, “What’s touching it?” Of course, I realize that a rope is touching it. The rope’s only pull. The direction is along the rope, so it has to be a force to the right. I always label forces caused by ropes with a T for tension.
Then there’s also contact with the surface of the table. That’s a normal force. The surface is pushing up to counter the weight of the object. The object feels gravity pulling it down and the table pushing up to balance that.
Okay. The analysis then is that the motion of this object has to be consistent with Newton’s Second Law. That means Ax has to be equal to the sum of the horizontal forces and Ay has to be a result of the y forces. And I’m going to jot down here that I’m going to go with the standard axis definition we use often.
As we’ve always done, the name of the game is picture the acceleration. We know that has to be to the inside of the corner, so it has to point in the same direction as T.
What are all the forces on the object? And those forces have to add up to and be consistent with a. That’s what these equations mathematically guarantee us. The y version of Newton’s Second Law, it defines or tells us that Ay is 0. I have only a horizontal acceleration, so F n has to balance m g. That’s what this equation will tell us. 0 has to equal plus F n minus m g. The m, remember, just drops out when the left side is 0.
For this circular motion, all the action so to speak, the acceleration is happening in the horizontal direction. I have an Ax. It points to the right. It would end up being a positive number. It’s caused by all of the horizontal forces.
Well, I only have one horizontal force. That’s the tension. My numerator then becomes plus T. It’s the only force. And I still have to divide by m. So this is Fnet.
It’s the tension from the rope. This you could refer to if you wanted to as the centripetal force. I would not, though. I think it’s confusing to do that. I would use the terminology that my acceleration for this object that’s following a circular path points to the center of the corner. And the force responsible for that is tension in the line.
That’s emphasized again by my last bullet here. I say it repeatedly, because it’s very common to get confused the way books describe this. Every force that’s causing my centripetal acceleration, my acceleration because an object is following a circular path has to be caused by a real identifiable agent. There’s no magical centripetal force.
It’s either a combination of forces, perhaps gravity and a normal force, or perhaps it’s only one force as my problem above here. It’s a rope. That rope is literally pulling that object around and keeping it following that circular path. There’s a pull on that object.
So here I have it written down for you to summarize it. In that example above, the tension force is what’s causing the acceleration. Some circles, some people would like to call tension the centripetal force. I wouldn’t use this language. I think it’s confusing. It soon has students thinking that they need to just draw a centripetal force as some extra new force on their free body diagrams. Not true.
Okay. I want you to think about the consequences of what we’ve just discussed as we watch this movie. Let’s match what I’ve been talking about to real life.
Here’s an amusing movie of a person that’s just going to roll a bowling ball across the floor. The person standing here with the mallet is going to hit this ball and do whatever it takes to try to make this ball follow a circular path. Let’s watch this carefully.
Okay. The ball is the — the bowling ball is the object that we’re focused on. And we’re trying to analyze what’s the direction of the force that causes the motion that we see. He’s trying to get the ball to follow a circle. What direction does he ideally have to hit the ball in order to get it to follow a perfect circle? Too bad there’s not audio on this. I bet he’s breathing pretty hard by the end.
So watch that a couple of times and be thinking about the forces on that ball during the motion and can you get a sense of what direction he needs to be hitting that ball in order to get it to follow a circular path?
Here’s a couple of quiz questions for you about that.
First question. It’s the bowling ball that we’re watching in this what ideally would be circular motion. He’s trying to get it to follow a circular path. Question three. What’s the direct force that’s responsible for the motion of that ball? That path that you’re seeing it follow on the floor?
Question four, then, is in what direction does that force have to be applied in order to get the bowling ball to follow a perfect circular path. Watch the movie again and anticipate if you had that mallet, how does he need to be hitting it to get it to follow a circular path?
Parts of the movie — it is following a better circular arc than in other parts. What’s the difference in the direction he’s hitting the ball in those two cases?
Well, that’s my warmup to Chapter 5. And this brings us then to the specifics of “Uniform Circular Motion,” the title of this chapter.
Uniform circular motion is nothing more than a simple — in fact, the simplest kind of a curved trajectory that you can imagine. The “uniform” here refers to speed. It’s at a uniform, constant speed following a circle.
Okay? Quiz question five. Keep you practicing motion diagrams and picturing this motion. You’ve got two choices here of motion diagrams. Which is a better representation or which is a representation of uniform circular motion as defined? What does it mean physically?
The next two quiz questions have you thinking about the consequence on the acceleration. We have to have acceleration if we’re following a curved path. Therefore uniform circular motion. I must have acceleration.
Take the time and go through the exercise of that vector process and sketching to figure out the direction and the approximate size of our delta-v. Consider it at this top point, so this is your initial velocity. This is the velocity following. What’s the direction of delta-v between the two? Therefore the direction and rough size of a between the two.
Question six. First just tell me the direction. At this point, does a point up, down, is it 0 or is it going to be some dramatically different angle than up or down?
So in question seven, then, I want you to contrast these two scenarios. The only thing I’ve changed between question six and question seven is the speed the object’s going around the circle. So the time interval is the same, but notice the velocity vectors are larger. My average velocity between these points is now larger.
Go through the same exercise of considering this initial velocity and this later velocity. Figure out what the delta v is. Therefore the direction of a and approximately the size of delta-v and therefore the size of a. And answer the question how is a different now? My goal here is to get you to think about the connection between a, the magnitude of this center pointing acceleration and the speed that the object’s going around the circle with.
Next thing I want to do is show you the equations that relate the value of the acceleration to the other physical characteristics. In other words, the meters per second squared of this acceleration isn’t independent. When something is following a circular orbit, the magnitude of this acceleration depends on the speed and it depends on the radius of the circle, how tight this circle is.
So here are the equations. They’re derived for you in the book if you want to see the mathematical logic behind why they have to be what they are. In this lecture, though, let me just emphasize what the equation says and what these variables represent.
The first one here points out this connection I just alluded to. The magnitude of this acceleration is not independent. It’s equal to, it’s determined by the speed that the object has as it moves around that circular path.
Notice that the speed has to be squared in order to get the proper value of the acceleration. It also depends on the radius. This is, you imagine, fitting a circle to this path. And what’s the radius of that circle?
So think for a minute about how these variables affect the acceleration. If I increase the velocity — in fact I have to take that larger number and square it. That makes the acceleration then much larger.
Notice that the radius is in the denominator, though. If I have a bigger denominator, this value, this fraction gets smaller. My acceleration decreases. So objects in a tight circle with small radius have large acceleration. Big radius circular orbits, the acceleration is smaller.
Let me note here for you that this is the real acceleration. Meaning it reflects the real delta-v, the change in velocity of that motion. Always points to the center of the corner or inside the turn. And let me note that this is what the book calls “a sub c.”
I don’t like that notation for the reasons I’ve pointed out. It coaxes you into thinking it’s some new kind of acceleration, which it isn’t. I probably will not use the variable a sub c as I walk you through examples like the book does. I’m going to stick with the variable a. It’s the same acceleration we’ve been dealing with.
Okay. Key equation. Let me highlight it for you.
Think of this as a kinematic relationship for circular motion. It’s a physical connection between the size of the circular orbit, the speed and the acceleration.
There’s one more key kinematic-like relationship that will help you out. For circular motion, the velocity — let me say that better or more correctly. The speed of this circular path with the object following a circular path is also related to some physical quantities.
So again, this is my speed. The speed is related to the circumference of the circle. Remember that a circumference of a circle is 2 times pi times r. So this is the distance around once.
Distance divided by time gives you speed. So T is the time to go around once. And that’s called the period. It has units of time, so seconds would be the SI unit we’d use for period.
So this physically I hope makes good sense to you. The speed as an object goes around in this uniform circular motion. I can calculate the number for that. Speed is distance divided by time. Well, if I consider one path around the circle, that distance and I divide it by the time it takes to go around once, that ratio will be my uniform speed.
This is another important kinematic-like relationship for you, connecting time around once, the distance traveled around once to the speed.
Okay. I’m going to do a couple of good examples here for you. But first let me just give you the big picture. What’s our strategy to solving problems for uniform circular motion? Not a lot’s different now. We’re going to follow the same steps we’ve done before. We’re applying Newton’s Second Law to this object that’s undergoing this circular motion.
From the picture of our tree analogy here, where we’re starting with basic equations and then we’re learning how to logically problem solve out to the solution for a specific problem, I have my collections of leaves or problems so to speak, which I would call “kinematic problems,” just describing the motion.
And we know about the kinematic basic equations that you always start with to solve those problems.
Then what’s highlighted in pink is a whole category that we call “dynamics,” which is applying Newton’s Second Law to a problem. It’s saying how the force or the forces impact the motion. And of course, that’s through changing the acceleration.
What we’re doing now with uniform circular motion is really just a subset of this group of problems that we would call “dynamics.” It’s just a special case. We’re still going to apply Newton’s Second Law, but we’re applying it to a certain kind of motion, which is called “uniform circular motion.”
And our starting point is still the acceleration for our particular problem is caused by adding up the forces, divided by mass. But now for uniform circular motion, you have some extra kinematic things to keep in mind. You’re never in equilibrium. You always have acceleration. And the value of that’s not unique. It is always related to the speed squared divided by the radius.
And you also may need or it may be useful to know that that speed you can express as the circumference of that trajectory divided by what’s called the “period,” the time to go around once.
These are just extra relationships you can bring in to help you solve the problem.
What I have next is just a copy of the problem solving steps. You can print out a copy of this through your lecture blank or your equations.
So what I have next here is a copy of our same problem solving steps. I’ve just modified it slightly to add this extra information. The idea is essentially the same, though. We’re using Newton’s Second Law to understand the acceleration of an object.
The steps just help us focus on what is a? What’s the left side of the equation for this object? What’s the right side of the equation? What are the forces on the object? Then dividing by the object’s mass. And I know that I break that equation up into its two component equations.
So all of our steps, one through seven, are completely the same. The only thing I’ve added or the difference is that the bottom of this page now I’ve put all of our other considerations we need to have in mind when we apply Newton’s Second Law.
Before, remember, I highlighted that if we have to consider multiple objects, this is really the Third Law. Information that we can bring in to solve the problem. We have to apply our Second Law, the a equals F over m to each object, but then we can connect forces through this, Newton’s Third Law. The values of these forces and directions. And then these multiple objects usually share the same acceleration, so we can also equate or draw a conclusion about the a’s that appear in the equation for each object.
The other thing now I’ve added is some comments for circular motion. Said another way, any object that’s rounding a corner.
Here’s the additional things you have to keep in mind.
You need to read the problem carefully and understand what snapshot you want to consider. Usually it’s best to imagine viewing that and picturing the forces and their direction as viewed from the side, because this will ensure that you see the forces that you need to consider.
Remember then these extra kinematic pieces of information. The acceleration always points to the inside of the turn. It’s value is not arbitrary. It’s set related to the speed and the radius. That speed’s not arbitrary. It’s related to the circumference and the period.