https://youtu.be/MpbZns9KsRQ
PHYS 1101: Lecture Eleven, Part Two
. . . quiz for this first problem to describe that to me.
Under the “Where Were We” section, let me give you an overview of Newton’s first and second law, which is what we’ve learned so far. At the heart of motion, I think, your first instinct, a person’s first instinct, is to ask the question, “Well, what causes motion?” and then the follow-up question to that seems to be to make a distinction between leaving an object versus something interacting with it and changing, or impacting, the motion.
So, if you ask yourself, this black and white difference, if you left an object alone versus not leaving it alone, what would it do? What’s its natural state? That’s at the heart of Newton’s first law.
It was revolutionary for him to point out that leaving an object alone means not that it comes to rest, which is what a lot of people would guess, but rather that the motion won’t change. If something’s stationary, it will stay put if left alone. If something’s moving along at a constant pace, in a straight line, constant velocity, that’ll continue if it’s left alone.
So leaving it alone simply maintains the current state of motion, whatever that is, but there’s a subtlety to it, that leaving it alone doesn’t necessarily mean that nothing can be touching it. It just means that there has to be no net effect. In other words, things can be touching it, but, as long as they balance and they counteract each other, that’s equivalent to really leaving the object alone. So that’s the first law. Fundamental: leaving something alone, an object doesn’t come to rest, but its motion doesn’t change.
The follow on to that then is if there is some net interaction, there is somebody pushing an object more to the right than something countering the push to the left, the motional change. And what does that mean? A change in motion means a change in velocity and we call this a net force.
I’ve got two examples here for you, where, for example, sliding a book across a table, we know that that book eventually will slow down and come to a stop or a person accelerating to the right speeding up, velocity vector’s getting larger.
From just looking at the motion, I know, because I’m speeding up or slowing down, that I have a delta-v every second. I have an acceleration. Therefore, Newton would tell us that, whatever the forces are on this book, there has to be a net force that points in the same direction as the acceleration is. That net force that’s causing the motion, not the motion itself, but the motion to change, it’s causing the acceleration.
So, by analog over here, if something is speeding up to the right, its acceleration points to the right, I have to have a net force to the right that causes this velocity to increase. Newton’s second law is what we’re going to focus on for a majority of this class and it’s going to be the interpretation of this equation, interpretation of the forces, that we’re going to spend a lot of time on and a lot of problems are going to focus on it.
Here it is mathematically, what Newton would tell us. This equation says that the vector a which I know I can physically think of as a vector delta-v, it’s the little change in the velocity vector per second. This vector is equivalent, it’s caused by, it’s equal to the sum of forces. That’s the sum of all forces that are on this object, but then it’s divided by the mass of that object.
Okay, the acceleration, that’s the real acceleration I would picture from a motion diagram for that object. If it’s speeding up, a is in the same direction as v, if it’s slowing down, these have to be opposite and I know, if it’s a curved, two-dimensional motion, acceleration is at a different angle than the velocity. That’s what’ll cause the velocity to change and to curve to follow that motion.
Okay, on the right side, let me say a few more things. This sum of forces, what gets substituted in for everything highlighted in blue here, is a vector sum of every force on the object. Common mistake is to forget a force, or to, if you forget any forces on the right, then this equation won’t correctly predict the real acceleration the object’s going to undergo.
This sum of vectors in the top here, in the numerator, they’re a sum of vectors so I will remind you that you have the tool of the tail to tip addition of these vectors. That’s an excellent way to have a visual picture of what F net is. That’s what we call the resultant of this sum, is F net.
We’re going to learn a little later that the name we want to emphasize later when we do the mathematics is the component description of adding these vectors together. That means focusing on the x components of all the forces add together to determine the x component of the acceleration. Likewise, the y component of these forces add together to determine the y component of the acceleration.
And then this mass in the denominator… the mass of an object is different from the weight. It’s related to that. I’ll explain that a little more later. Think of the mass as being an intrinsic property of this object that represents, in some sense, it represents the total number of molecules that just make up that object. It’s just the total mass of it.
The total number of molecules for this object would be the same whether it’s out in space or it doesn’t feel any weight, it’s weightless or on the surface of some planet. This is just an intrinsic property of this object. Think of it as number of molecules. It’s a measure of the mass. It’s going to be so many kilograms and we’ll call that inertia of the object.
As a last summary of this equation, I’m going to make one more point again: these forces add up to give me a net force. The net force causes, not velocity, but acceleration so it causes a velocity to change.
I’m going to write that again just for added emphasis: Note. I’m going to write that as, let me just write what that equation would be, that most people instinctively want to first try to interpret this equation to read, that your eye is drawn to the velocity and there’s a strong intuition, a strong tendency to say that it’s this velocity that’s directly caused by some interaction, some forces on that object.
This is the logical thinking that goes along with the conclusion that an object if left alone, if this was zero, would come to rest. The velocity would be zero and we know that’s not true. If an object is left alone, its acceleration is zero so I’m going to draw a huge red x through this.
Forces don’t cause velocity, they cause delta-velocities. They cause velocities to change.
The last thing I want to summarize from the previous lecture are these steps for identifying forces. You have to remember, in physics, force is going to mean something very specific. In our class, every force is going to be a real contact force. It has to be because a surface, an object, a person, something is directly touching that object.
The only exception to that is going to be the one long range force we’re going to cover in this class and that’s the force due to gravity. We’re going to always call that w, that stands for the weight. We’ll discuss the magnitude of this force in a minute but the weight, the force due to gravity, is what our sensation of weight comes from and that’s why it’s going to be labeled w.
Okay, forces are always a push or a pull, and you always can point to or identify the agent or the object that’s responsible for this force that’s being put on an object. My cartoon of how best to picture and be sure you don’t leave any of these forces out, is to draw in red, focus on the one object you’re interested in.
Then, imagine a circle right around that object. Then, follow around the edges of the circle; anything that’s in contact with that object, that breaks the circle, touches the object, there’s a force at that point due to that contact.
So, for example, this person was standing on the floor or this person was pulling on a rope. As you went around this circle, you would say, “Oooh, rope’s crossing the circle, touching the object, there’s a force.” Ropes, we’re going to learn, always pull, has to be a force up. You continue on around the object, nothing touching here, nothing touching, “oooh a surface.” Surfaces, we’re going to learn, always push. There’s a force up from the object, from the surface.
After you do this contact test of what’s in contact with the object or before; course you do need to always include this long range force, the force due to gravity.
Here’s a couple of quiz questions to emphasize this connection that F’s, forces cause acceleration, not velocity. So, when you’re trying to work with any of these problems and there’s a question about forces and you’re having to think about Newton’s second law, picture the acceleration and the forces and go back and forth. Be sure that the force picture that you have is consistent with the acceleration picture that you have. That will help you when answering those kinds of questions.
The next two quiz questions are reading quiz questions and, as is often the case, I’m emphasizing memorization of symbols and variables as you read the text. What will the symbol small w with an arrow over it represent? What will the symbol of small f with a little k subscript and an arrow represent?