https://youtu.be/Mvbzm4KkjEQ
PHYS 1101: Lecture Ten, Part Six
Now let me show you the mathematics of it, the translation of that into mathematics. The one we focus on, spend most of our time on is really the second law. That’s the one that we can leverage to solve problems. Remember the second law said if there is a net force on an object it causes the velocity to change. That means the object is not left alone. Something is interacting with it. There’s a net force and that causes the acceleration, velocity to change.
Here’s the equation that says that. The acceleration for some object results from a sum of forces divided by the mass of this object. Let me specify these more carefully. First point, in this equation the a that you see is the real acceleration vector that we’ve been identifying in these motion diagrams. It’s the real delta-v that you see as the motion progresses. It’s the acceleration for this specific object whose motion diagram you’ve drawn. So note, we’ve got a focus on one object.
On the right side of this equation it says that this acceleration is caused by the net force divided by the mass. This numerator, that funny symbol there and then the f with the arrow over it, that means the sum… I’m going to be more careful. The vector sum of all forces on the object. That symbol is called Sigma. It’s a Greek symbol. It’s a capital letter Sigma and it’s just used in mathematics to symbolize summation or sum instead of having to write out sum.
Okay, this is the vector sum of all forces on the object so let me sketch what that means. Let’s put a spot here and imagine that this is the object and we’re just looking at a snapshot in time, and at that instant we’re told that there is a force to the right on that object and perhaps there’s a pull downward on that object, and that a careful analysis has been done and these are the only two forces acting on that object. In order to properly determine what this numerator is I have to do the vector sum. That’s the tail to tip addition of these two vectors.
How do I do that? I’m going to write here that that vector sum, I’m going to call it F net, the resultant force is equal to the vector sum of F1 plus F2. So I’m going to do tail to tip. I’m going to draw F1. Then to the tip of that I’m going to draw my next vector, F2. Then the vector sum goes from the initial tail to the final tip. That’s F net. So the equivalent of having these two forces on this object is having this single force down and to the right on the object, or F net. That’s the equivalent, everything here in the numerator.
Okay, this last term. What’s that m? The m there has to be the mass of this specific object whose motion it is that you’re analyzing. The mass of that object, and I think I’ve mentioned this before, mass always has units of kilograms. Just like this. F net, the force here will have units of Newtons, kilograms meter per second squared.
So down below I’ve sketched a little flow chart here to be sure you understand this connection. I know I’m repeating myself but sometimes saying it in a different way helps. It really all starts with a picture of what is the net force on an object, or do they all balance? Perhaps the net force is zero.
Let’s consider that I do have a net force though. This vector addition does lead to some net effect. The net force then causes the acceleration and that’s what this equation says. The acceleration is caused by this net force. That acceleration we know means that the velocity changes. I have a delta-v. So with this f net, the direction of that force tells me the direction of my acceleration vector. This is what’s called a vector equation. The direction then of the vector on the left has to match the direction information on the right which only can come from the forces. Those are the only vectors on the right. The magnitude or the value of this vector has to come from the magnitude or value on the right-hand side.
So when I draw these arrows here, what’s really critical to drawing a is to be sure and capture the right direction for the acceleration. The magnitude will come from doing the real numbers and the mathematics of the right hand side. With your little arrow to just capture the acceleration, just get the direction right. This acceleration arrow we know is the same direction as our delta-v vector.
So I’m going to go over here to the right and say what is the next velocity? It changes by this amount. How do I do that again? At this next point I’m going to redraw v0, but now I know that I have to add v0 plus delta-v gets me my next velocity vector. So rather than this object continuing straight, because there’s a net force on it it accelerates. It causes the velocity to change by this amount, and so that is our next velocity. Here’s point 1. We bold these points. Here’s 1, here’s spot 2 in our motion diagram, and here’s spot 3. This object is going to start curving up because of this net force.
Here’s said in words what I summarized above. When you think of Newton’s second law in its equation form, this acceleration vector gets its acceleration from the direction of F net and it gets its value from the value of the right-hand side. That value will be the magnitude of F net in units of Newtons kilograms meters per second squared divided by the value of m. M is just a scalar. The mass of an object doesn’t have any direction information associated with it, so this calculation will set the value of the right side and therefore that’s the magnitude or the value of the left side. Let me change my highlight here to blue. That value is calculated as I indicate there.
So let’s look at this equation and let’s just assume it’s in one dimension, meaning I have a force maybe that’s pushing to the right, only one force and so the object is going to accelerate to the right. In 1D, this equation would become that the acceleration is equal to that one force divided by the mass.
I just tidied this up a bit to make it more specific. Let’s do one-dimension and only one force applied to it. This equation says that the magnitude of a is equal to the magnitude of the force divided by m. If we put in the scalar components here for F, meaning perhaps we said the force was to the left so we made this minus 10 Newtons, then a would be negative and we would then know that a, the scalar component points to the left. That vector points to the left.
The next two questions have you thinking about this mathematical relationship. The value of a and how it depends on F and m. What happens if you double the force? How will the acceleration change? What if you double the mass? How does the acceleration change? So this is an exercise in translating from the equation back to real life.
My first set of questions here for you is that a constant force applied to an object causes 5 meters per second squared acceleration. The same force applied to object B causes a different acceleration. And then object C the same force results in an even different acceleration. So if A is changing, the force is staying the same, I have to have different values of m for these objects.
So question 12, which has the largest mass? A, B or C?
And then question 13, which has the smallest mass?
Question 14 and 15 also applies to a simple one-dimensional interpretation of Newton’s equation. A equals F over m. Now it asks you that given that a constant force causes an object to accelerate with 10 meters per second squared, what happens to the acceleration if only the force is doubled? What will the acceleration be if the mass is doubled?
A good approach to solving problems like that is to literally picture. I know a was 10 meters per second squared for a certain force and mass. If I then substitute in 2F, how is that going to impact a? How does that factor of 2 here cause the value to change? Pick some numbers. Pick force values and pick a mass and actually change the force to a factor of 2 and see how a changes if that helps you.
The next few quiz questions have you thinking carefully about what the F net will be when you look at a particular scenario for some forces. Again, you want to think of this as not that the object is stationary, that this is a motion diagram, but this black dot just represents that object. Think of it as an instant. It’s the object and these are the forces on it.
So looking at this scenario I’ve got two forces and you’re trying to determine if the object is accelerating or not. At the heart of it is the fundamental equation, Newton’s second law. You need to think about the vector sum, the tail-to-tip addition of these forces. If this is not 0 then the object is accelerating.
Question 17, it’s the same question. I’ve just changed the drawing slightly, so the forces are a little different.
Question 18 has you comparing two objects. Scenario 1, I have no forces on this object so for sure F net is 0. Object two, you’re looking at this force combination and I ask you how do the accelerations compare for those two objects?
Question 19. Again, a single snapshot. Here are my two forces. Can you tell if this object is speeding up, slowing down, not accelerating, or maybe you can’t tell from that picture. This one deserves careful thought. I’m going to give you plus 4 versus plus 1 on this question. Bring this one up in the discussion board.
Question 20. Bring this one up in the discussion board. Here I’ll give you plus 5 or plus 1. The snapshot shows you the two forces on the object and I ask you can you tell if this object is moving? Is it moving? Yes, no, or can’t tell.
Question 21, look at this combination. At an instant later, is this object moving? Yes, no, or can’t tell. I had a lot of quiz questions here. I apologize for that. But if you take the time to think these through it really will serve you well in the lectures to come. I only have I think two or three more.
Question 22. I’ve got three forces here shown on this object. Which arrow best shows F net? This is an exercise of going through the tail-to-tip addition but now applied to forces. Same thing we did in Chapter 1. We need that skill now, thinking about the vector sum of forces.
Question 23, same idea. I need to figure out F net, the tail-to-tip addition given this force combination. I know then from Newton’s second law that the acceleration direction is a consequence of the direction of this f net. The sum, the tail-to-tip addition I’m going to call the resultant vector F net. What does the resultant vector look like for F1 and F2? Given that direction, that’s the same direction for a so which is your best choice off to the right?
Thank you for your patience. That brings us to the end of Lecture 10. I look forward to your discussion on the discussion board and helping out.