https://youtu.be/7AIeKoVmEUc
PHYS 1101: Lecture Thirteen, Part Two
I’m going to start our new material by motivating it with a problem. Let’s say you were given a scenario as shown, and you had to figure out what tension do you have to pull on this rope with in order for this blue mass to accelerate to the right with a certain value. Meaning, every second, my velocity increases by 4 meters per second.
So, maybe it starts from rest. A second later, it’s got to be going 4 meters per second to the right. The next second, 8 meters per second, et cetera. The green is a rope that connects these two blocks together over this pulley. Then, on the left here for the top block is a rope that I’m pulling on, leading to this tension force on mass one, the purple block.
Okay. Many problems you’d have to deal with look like this. They’re not as simple as scenarios we’ve considered so far. When you ask yourself problem solving step number one, what’s the object, it’s not obvious. I’ve got really two objects here. Can I treat them as a combined object, as though they were attached? Should I start putting the forces on all of these blocks? On that entire object, or the entire shape here, as though it were one object? The answer is no. You can’t treat it as one object.
Definitely, if ever you have motion between the two objects or two objects in a problem, or if you’re asked about the nature of a force that would be at the contact between these two objects, you can’t treat them as a single object. You’re going to have to separate it out into two objects. So, I’ve written that out for you.
Remember those problem solving steps. The equation, Newton’s Second Law, acceleration is equal to forces on an object divided by the object’s mass. It only applies to a single object. If you have motion between objects, you’re interested in forces between objects. You have to break it up and apply this equation, Newton’s Second Law, to each of those objects independent.
Okay. So, we can’t treat them as one object. I have to treat them separately. So, let me begin to walk you through some of the steps. We tried to apply Newton’s Second Law to each of those objects.
Let’s start with the purple box, and let’s draw our free bodied diagram for that object. Let’s sketch the forces that we know are just on the purple box. Well, I know I have the weight. This is the force of gravity on the purple object. I’m going to call that weight one. Then, what else do I have? Now, I need to consider going around the surface of this box and ask what’s in contact with it. Let’s start here at the side.
Well, I have a rope touching the side of the box. That’s going to put a tension force. I’m going to call that t2 because it’s not obvious at first that it’s equal to the value of t1. If it is, my analysis will tell me that. I’m just going to call it t2 for now.
Continuing around the surface of the purple box, nothing touching it on top. On the left side, a rope is pulling and attached to the surface. It’s putting a force t2, I’m sorry, t1 on the box. Now, I continue around to the bottom surface.
Now, remember whenever you run into a surface, there’s two forces you need to consider. You definitely have a normal force. I’m going to call that fn, and I’m going to put a 1 there too, just to remind myself this is for the purple box.
A surface can also have roughness, and I can have a frictional force. I’ll have that if there is motion at this interface between these two surfaces, or if there’s a tendency to motion. Well, I’m told in the problem that the tension here is large enough that it’s pulling the box and accelerating the purple one to the left and the blue one to the right. So, there definitely is motion between those two surfaces.
The friction force is in the direction to oppose that motion. The purple box is moving to the left. So, the frictional force has to be to the right. It would be kinetic friction, because there’s definite sliding between those two surfaces.
Let’s do a quick visual and see if these forces make sense, because I’ve exhausted all possibilities I can think of now. I’ve covered everything that’s in contact, and at every contact point, I’ve considered the nature of the forces I can have there.
Vertically, does this look good? Yeah. It does look good to me, because the purple block is going to accelerate to the left. There’s no vertical component to that acceleration. So, everything vertical on my free body diagram has to balance. It looks like I did a pretty good job of that.
Okay. Now, right left, the visual addition of these forces has to be consistent with my acceleration to the left. Fnet has to point to the left. That means this force, when I subtract these two, better be larger. I better end up with a net force to the left.
It’s not obvious that would be the case. So, I’m going to go in and make the tension a little larger. So, I for sure have a visual that there’s more force to the left than the addition of all the forces to the right.
Now, let’s do the blue box real quick. What do I have? Well, I’m going to have weight. I’m going to have the force due to gravity on the blue block. Now, let’s go around its surface. Let’s start here on the left. I have a rope. Ropes can only pull. That rope is providing tension. That is t2.
The rope that goes around this pulley is going to have the same springiness, the same tension. If you were to pluck it here and pluck it here, it would have the same resistance to that. That means the tension would be the same everywhere on this rope. So, also, that’s the tension that it’s pulling with against the object. I’ve called that t2.
I continue around here. I run into the top of the box being in contact with the purple block. What forces are associated with that contact? Well, it’s a surface pushing down. So, I know I have a downward force. Let’s call that the force from block one pushing down. It’s a surface with potential motion at that interface between these two objects. So, I may have friction in addition to a normal force.
Okay. The direction of motion. What direction would that force be? Well, the blue block is moving to the right. So, the frictional force on this object has to oppose that, and it has to be to the left. That’s a kinetic friction force.
Now, I move around to the left-hand side. No contact. No ropes. Nothing there. Then, I hit the bottom surface. I know I have a normal force. I’m going to call this the normal force on block two. There’s roughness at this surface. I have sliding between the blue block and the ground. The block is moving to the right. So again, that frictional force has to be to the left.
My acceleration for this object is to the right. So, the forces better at least give me a visual addition with something left over to the right. Let’s first check if up/down balances. It looks like I didn’t draw that well, that there’s a net force down, which isn’t right. That gives me a better balance now for the vertical acceleration having to be zero.
Let’s look right/left. Looks like I’ve got more forces to the left. So, probably, I was a bit heavy handed with these two forces here. I’m going to make them a little smaller. Where everything right/left, now, does add up to a net force to the right, and so is consistent with this picture.
Okay. This is a complicated problem. I’ve got a lot of forces here. Some of these forces, I know, are because of the interaction between the purple block and the blue block. You can, perhaps, intuitively agree that I also have some interaction between these two blocks through this rope or this line.
These forces that are interactions between these two objects, what more can you say about them? That’s at the heart of Newton’s Third Law. So, for a problem like this, these two objects are moving in opposite directions. There definitely is interaction between the two objects. We have to apply Newton’s Second Law to each object independently.
Newton’s Third Law is going to give us some very useful information about the nature of the interaction between these objects, how large these forces are. At this contact point, how did these forces relate? That’s going to help us draw some logical conclusions about the size of some of these forces, and therefore, give us more information to solve the problem.