https://youtu.be/bHF5gXF3S8U
PHYS 1101: Lecture Thirteen, Part Four
So, I want to say something about including this in your problem solving strategy. What I’ve done is taken our problem solving steps, and I’ve just modified that page slightly. At the top, I’ve included this reminder that, rigorously, Newton’s Second Law and these steps, which have you focusing on the acceleration of that object and the forces on that object, only apply to a single object. When do you have to invoke Newton’s Third Law? And when are you going to have to include multiple objects? The box I’ve added in pink summarizes that for you.
I really think of it as two criteria, if one or both is true. If you have motion between the two objects… Yeah. If you have motion between two objects or more in a problem, you need to apply f equals ma to both of those objects to determine the logical consequence of, perhaps, the tension force or the frictional force between the two surfaces, et cetera.
You then, also, have to now consider two things. Because you’ve got multiple objects that are interacting, either through a rope, or they’re in direct contact with each other, you have Third Law pairs. They have to have equal magnitude, opposite direction. Invoking that to equate a number that you get, say, from solving Newton’s Second Law for one object, will give you a value then for something you need for the other object. So, equating the value of those forces, opposite direction, is going to be key.
The other thing you need to appreciate and recognize is that you do have some information about how the accelerations of the two objects are related. What does A look like for both of these objects that’s on the left side?
For example, at the very top of our lecture, the beginning of our lecture, we looked at this scenario where we’re pulling on a rope on the purple box. Physically, that’s going to, through this rope, lead to a pull on the blue box.
How do the accelerations of these two objects compare? For sure, the acceleration for the blue box is to the right. We were told that and given the value for it. Just in your mind, picturing, watching this, you can picture that this object is definitely moving to the left.
One thing to keep in mind as you consider the orientation or how these accelerations compare is how are they connected. In this scenario, we’re connected through a rope. If that rope stretches, then it could be that this purple object moves a different distance, say, in a second, than the blue object. For sure then, the velocities would be different, and likely, the accelerations are different. That’s if this rope stretches.
If you’re not given any information about how it stretches, and usually, you won’t be, the idea is that this rope doesn’t stretch. It stays nice and tight. So, if this blue box moves over a millimeter, the purple box moves to the left a millimeter. If the blue box does that every second, so will the purple box, one millimeter per second to the left.
It logically continues on through to the acceleration. If the velocity of this blue box increases to the right by four meters per second every second, if this rope doesn’t stretch, the purple box has to accelerate to the left by the same amount. Every second, the velocity has to increase by four meters per second.
So, for this scenario, here’s how they compare. The accelerations are the same magnitude, but the directions for this configuration are dead opposite. Blue box accelerates to the right, purple box to the left.
So, when I went to solve Newton’s Second Law, if I chose to use the standard positive x coordinate system for both objects, when I went to apply my problem solving steps to each of these, the sign of my acceleration would be different. For object one, the purple box, my acceleration would have to have a negative value, minus 4 meters per second squared. That’s what would be consistent with the equations for this object, based on that coordinate system. For the blue object, I have to have a positive 4 meters per second squared as I do the mathematics and work the equations for the blue box, given this coordinate system.
You don’t have to have the same coordinate system for the two objects. You do have to choose a coordinate system for each object, and then follow all your problem solving steps through. Based on that coordinate system, you have to be consistent in that regard. Whatever you choose for one object, you use it consistently through the entire equations for that object. Then similarly, whatever coordinate system you use for your other object, follow it through. I just solved the problem.
So, that’s all I’ve done to help us solve problems that involve multiple objects, is simply to add this comment to our current problem solving steps. We’re going to apply these steps always to a single object, but if we conclude that we do have to consider multiple objects, each object gets this set of considerations, independent.
But then, there is some conclusions we can draw about the motion of the two, the similarities that they have. What is related between the two? That’s going to be the consideration of Newton’s Third Law, the action/reaction pairs, and the acceleration comparison.
Accelerations, as long as a rope isn’t stretching or shrinking, and I have yet to give a problem to my 11-01 students where that’s the case, then the magnitude of A is always the same, but the sign of the scalar component that you use for each object might be different, based on the coordinate system that you choose for each of those objects.
Before I do our first example, I want to go back up to the beginning of the lecture, where I first suggested this idea that we have to consider these two objects separately, and make some general points for you.
Let me add here that you must consider objects separately if… I’m going to put two bullets here. If two cases are true. Definitely, if there’s motion between these objects, or the motion is in different directions for the two objects. The other bullet is to point out that if you’re asked about the nature of the interaction between objects, if you’re asked about or you need to consider the size of the friction force between these two objects, you need to break the objects up into two objects separately.
So, let me write here, “If you need to consider forces between the objects.” That’s the best way I can describe it to help you decide if you need to treat them as two separate objects or not.