https://youtu.be/WA0Kfnu2f2E
PHYS 1101: Lecture Twelve, Part Two
Here’s my summary for you of where we are to date, the summary of last lecture. Last lecture, we covered, primarily two things, I gave you an overview of the common forces that we are going to encounter from day to day objects that we consider, and then walked you through some problem solving steps for applying Newton’s Second Law.
Whenever we’re working a problem and there’s any mention whatsoever of forces, a tension, acceleration, connecting that to forces, our starting place is always going to be Newton’s Second Law. And, that means, applying these F equals ma steps that I suggest. Those steps do nothing more then help you slow down and build a picture of what needs to be plugged into this equation, which is the mathematical representation of Newton’s Second Law.
For example, the left side of that equation is the vector a, that’s the acceleration. That has to match your real physical picture from your motion diagram of what acceleration that object is actually undergoing. Is the velocity changing length, is it speeding up or slowing down, is the velocity changing direction, that also needs acceleration. What is that Δv, what is the direction of that vector a? What’s the left side? This left side has to be equivalent to, is caused by, is equal to, is a consequence of the effect of all the forces acting on that object.
The sigma, and the sigma means sum, this is a vector sum of all the forces on the object. We call that a quantity, everything highlighted in blue there, the net force.
I introduced a tool to you last lecture called the free body diagram to help you picture what needs to be plugged into the numerator here. What do you have to include as all the forces? You put a dot to represent the object, so this would be at that snapshot, at that instant. This dot represents the object, what are all the forces? You always have gravity acting on the object if it’s near a heavy, massive object, like a planet.
Then, in addition to that, you have to ask yourself, at that instant, what’s touching the object. Only things that are in direct contact can be exerting a force on it. Everything that’s touching the object, roughly how large is that push or pull, and likewise, what’s the direction of it. You want this free body diagram to visually match the vector acceleration picture that you have.
That means that the tail to tip edition of all of these vectors, the forces, those add up to give me a net force. The direction of my acceleration has to be the same as the direction of that net force. If the forces all cancel each other out, I have no net force, and so, I’ll have 0 acceleration. The denominator here, is the object’s mass.
Think of the mass as a property that’s intrinsic to that object that captures, literally, how many molecules make it up. Mass has units of kilograms. This is distinctly different from what we call the weight. The weight is the size of the force due to gravity acting on that mass. In day to day language, some would think of weight and mass as being interchangeable, but technically, we have to make that distinction. An object’s mass is the same regardless of where the object is, if it’s sitting near a massive planet, it’s out in space, or near the moon. The weight, however, the gravitational force because of this object’s mass is different if the objects near the planet Earth, out in space, or on the moon.
So, this is the vector equation, the resultant vector representation of Newton’s Second Law. When it comes down to doing the mathematics, we are going to, again, leverage using the one-dimensional versions of this equation, meaning we’re going to look at the horizontal components of the left side and the right side. And then we’re going to have a separate equation that represent the vertical components of the left side of the equation and the right side of the equation.
We’re going to do that because, to capture the direction of these vectors, once we’re working in one dimension, pure up-down and pure right-left, we can use the sign of each quantity to mathematically represent, or as a tool, the direction of these vectors. So, become comfortable with picturing the resultant vector interpretation of this equation compared to the component picture which says, you can think about the horizontal forces causing horizontal acceleration and at the same time any vertical components have to be the only things causing a vertical component to the acceleration.
In the last lecture, I also went over the main forces that we’re going to consider in this class. The kinds of forces that typical objects have to experience or do experience. The first one was the force due to gravity. Any object that finds itself next to a very large and feels the gravitational attraction, a real long range force that is in the direction towards the center of that big, massive object, we find ourselves near the surface of the planet Earth. It has a very large mass, the result of that is a long range force that we call the weight, which is significant.
It’s value, our weight, is given by our mass times the acceleration due to gravity. It’s also called the Earth’s surface gravity and has a value of 9.8 meters per second squared. In a later lecture, I’m going to derive this for you and show you the physics behind why that value and the weight is what it is. The direction of this force is always towards the center of the earth. And so, from the perspective of objects that we look at day to day, that’s always going to be down for us, whether we’re here or we’re in China.
The next three forces result from the interaction of this object with a surface. If an object is touching a surface at all, at that instant, because of surface contact, you have a normal force. The value of this normal force will be, as I say here, given by the a equals F over m prediction. What that means is, the value of FN will be what ever it takes to balance all other force components that are perpendicular to the surface.
I’ve come to that conclusion because the acceleration in the direction perpendicular to the surface, meaning, let me draw my object here. In this direction, perpendicular to the surface, the object can’t be accelerating, it’s not speeding up away from the surface or speeding up into the surface. In our day to day experience, this object, at most, is sliding along the surface. So, all forces in this direction, have to balance.
Whatever force is applied to this object, I know I have, for example, weight that will contribute, there could be other people pushing on this object, perhaps a rope pulling. Whatever the net effect of all that is, whatever’s left, to balance out and make the forces 0, that’s what the normal force will be. The normal force reflects the load that that surface experiences. Always perpendicular to a surface. Our surface, can only push away.
We can have two types of frictional forces. One’s called kinetic, and one’s static. We’re only going to have a frictional force if there is real sliding between the object and the surface it’s sitting against, that’s kinetic, there’s real motion between the surfaces. Or, there’s a tendency to motion, in which case I’ll have static friction.
The tendency for motion means, picture that the surface all of a sudden becomes perfectly slick, would the object slide? If the answer to that is yes, then I have to have a static friction force that must be keeping it in place and keeping that from happening. Static friction, you have to be careful of because there’s two regimes you need to consider.
The first regime is a case where, perhaps, the push on this object to try to get it to move is just very delicate, small. So the static friction is just as big as it needs to be to keep it from moving. If the force that’s trying to move the object gets large enough it will be able to overcome and to break all of the chemical bonds that are keeping those two surfaces in contact and keeping them static.
At that threshold our static friction is at it’s maximum value. It’s given by the product of a coefficient of static friction times the magnitude of the normal force. This coefficient is called μs, it’s a Greek letter, and it’s purely a measure of how rough these two surfaces are. It’s determined by the material properties of the object and the surface.
The normal force plays a role because, again, that is a measure of the load that this surface is supporting. That load, you can think of, also impacts how pushed together these two surfaces are, so to speak. It’s a measure of how many of these bonds are actually formed between the object and the surface, and, so, how hard I have to push to break all those bonds and get the thing to move.
Once it’s moving, the kinetic friction always has the value of a coefficient of kinetic friction times, again, the normal force. So, if you run into a surface as you’re going around your object, for sure you include a normal force and then you have to consider frictional forces.
The last one I put on the list is the tension. Any rope or wire on an object is usually there to support the object or to pull it. Tension always is a pull, a rope can only pull on the object, and the direction of that pull will always be in line with the rope. The magnitude of tension often is given in the problem or, again, based on what the acceleration is, looking at the other forces may then tell you what the tension has to be in order for that equation to be true.
Okay, on the right, I’ve got a blue, green, and a yellow bullet, and I want to emphasize which forces correspond to these categories. The first category I have here, blue, is that the force will have the same value for that object regardless of the size of the other forces. That’s only true for the force due to gravity. The magnitude always is what it is, that object has so many molecules, if it’s on the surface of the earth, the weight is always m times g.
The next category is what I’ll call a partially fixed value. This is the case for the two friction forces, the maximum value that the static friction can have and the value of the kinetic friction. I mean here that it’s partially fixed and that I have to multiply by some number, between 0 and 1. It’s just a fraction that represents how rough the two surfaces are. I have a kinetic version and a static version of that coefficient.
But, the magnitude isn’t just this fixed value, I have to also ask, how big is the load that the surface is supporting. This will reflect how many bonds have formed that I need to break. So, FN, I need to multiply by.
What I’m going to highlight in yellow are forces that, really can be anything, meaning that to say anything about that value, you want to take a careful look at what Newton’s Second Law, or what the balance of forces tells you that that force has to be. Something like the tension force may be given in a problem that, as I say. But, often, that tension will be whatever makes this equation consistent. That normal force will be whatever makes Newton’s Second Law consistent, etc.
Okay, we’re going to see problems where we continue to use and define and solve for values of these forces as we go on here. Let me move on to the last thing I wanted to review.
I have a copy here in these lecture notes of the problem solving steps. I won’t go through these in detail, but I did want to remind you, at the beginning that before you decide to use these problem solving steps, be sure it’s the right kind of problem, and that’s the right tool to apply to the problem. Here’s the rule of thumb, if there’s any mention of forces in the problem at all, there’s a discussion of tension, of weight, anything that’s a force, you will have to start with F equals ma because, fundamentally, those forces have to be consistent with the acceleration for that object. And that’s the mathematical scenario that describes that situation, that defines what those forces have to be. So, start with these problem solving steps, focus on these generic equations.
But, you’ll find some of these problems not only talk about the forces, but then they also have some kinematic flavor to them, meaning there’s a part of the problem that is about describing the motion. The connection is the acceleration. From our kinematic equations, and the kinematic information, like what the initial velocity is, the final velocity, the distance to slow down, etc., that kinematic information defines a real number and direction for a, and that’s the same a that is represented in Newton’s second law. And, so, that’s the connection.
It may ask you for the net force, given some kinetic information, so you first have to solve for a, and then get F net. Or perhaps, it gives you enough information for F net, you can then solve for a, and then, based on that a, you can, perhaps, determine how far the object goes before it comes to a stop, or something like that.
So, the reading assignment for this lecture was section 4-11. This section in the book discusses equilibrium applications and I want to just emphasize how my lectures here are departing a bit from the order that the book goes through, this application of Newton’s laws. I intentionally, through these lectures, are intermixing equilibrium applications with non-equilibrium applications, meaning that we’re looking at problems, in some cases, where the acceleration is 0, which is equilibrium, or we’re working with problems for which the acceleration is not 0, so it has some magnitude and direction off in an angle. And these are called non-equilibrium problems.
I’m mixing this up intentionally because I want to emphasize to you that it’s same physics, it’s coming from the same equation, namely that whatever acceleration we have is a result of the force scenario that’s on that object. It’s the same physics, the same equation, it’s just a question of whether or not the left side is 0, or not, whether these forces balance, or not. At any rate, the reading quiz still focuses on the specific section that I’ve assigned. So, these questions do have to do with, what does equilibrium mean. There’s question 4.
Here’s question 5, asking you more about what equilibrium means.