Hi, in this video we are going to learn about the Kinetics of the Particles, namely the force and the motion relationship. Before we start let's briefly review how the whole textbook consists of. The first part is about the Dynamics of Particles. And the second part is about the Dynamics of Rigid Bodies. Ultimately, as I mentioned before, we want to know what are the force and the motion relationship of the particle and the rigid body. But to do so, we have to know how we can formulate the acceleration. In a case for the particle and simple Cartesian coordinate, formulating the acceleration is somewhat easy. However, as we have studied before in Chapter 2, when you want to formulate the acceleration or the velocity in the rotational coordinate, or if you have many bodies, many linkages to generate the motion, sometimes it's kind of tricky and complicated. So usually before we learn kinetics of the system, we study about how we can formulate the acceleration in the velocity, which is kinematics. Now before we go through how we can formulate the F = ma, let's study the very important concept about how to figure out the force exerted on the system. We call it as a Free Body Diagram, it's a pictorial expression. How many forces are exerted on the body? We can get the idea from the name itself, the very first step to draw the free body diagram is actually free the body, okay? For example, if you want to draw the free body diagram of the robot arm of the Hubo, or the disc on the floor exerted by a stick, through the stick, the very first step is actually you should detach the body from all the other environment. Actually free the body, so free the arm, free the disc. And then note that all the contacts exert forces. Of course, there are forces exerted without contact, such as a field force, electrical field force, magnetic field force, or the gravitational field force. Other than that, all the contacts exert the forces. So for this robot arm, when we actually free the body, we can identify the contact, the stigma of the contact, here is the shoulder part, right? So since we are handling the 2D motion, we can add note that there are x directional, y directional contact force exerted to the body. And there is no other contact here, right? The arm is on the air, right? So the shoulder joint is the only contact that exert the forces, plus the gravity, the mg, okay? How about the disc? This one has a two contact, one through the stick and one on the floor. So at each contact point you have a horizontal and a vertical component of the contact forces. Sometimes it's obvious that the direction of the force. Here you apply the force through the stick, so the first direction is along the stick. So I just note it as it's that direction parallel to the stick here. So plus, there is mg applied to this disc, okay? This is the free body diagram, first detach the body, free the body from all the other environment. And just count how many contacts does this one have? And then each contact x and y components of the forces. Sometimes if the direction is obvious, just write the forces in that direction. And that's it for the free body diagram. So let's do the example. So this one has a two block connected by the two double, Pulley, attached to the spring and so on. The very first step is what? Okay, free the body, so I just blur the environment and highlight a body part as a red color. So how many contact does this block A have? Forget about those strings that connect to the first pulley and the second pulley, forget about that. Just keep focusing on this block, how many contacts? Two, one with the string, one with the surface, inclined surface, right? So two forces at each point, x and y, the normal force and the friction force. And this one is obvious, through the string, so the tension is along the string, and the mg, that's it. That's the free body diagram of the block A. How about block B, okay? Again forget about all the other structures and just keep focusing on the block B. How many contacts? Top and bottom, bottom with the spring, top with the string. So there are two forces, the direction, just consistent, so upward the tension T, and the spring force kx, plus mg, okay? Let's draw the free body diagram of the block C. How many contacts? One, two, three through the spring. The direction is sort of identified along the spring, and then one with the contact force. So contact has x and y components, no more force in the friction force, and then three spring force is kx, plus mg. That's the free body diagram of this block C. Now how about this three masses on the string? Let's focus on mass A, how many contacts? Top and bottom, so string on the top, string on the bottom. So you only have two forces, T1 and T2. We don't know if the T1 and T2 is similar yet. So just note that those two forces are different, and the mg, that's it. How about the block B, how many contact does this B have? Top and bottom, right? So you have a force top and bottom, so T2 and T3, the same string. So the T2 applied on the block A is the same as T2 here, and the mbg, as well. How about the C? C only have a one contact, upward, so there is a T3, the tension, and then mg and that's it. That's It for the free body diagram of this block A, B, C connected by the spring. Now let's do another examples. There is a mass connected to the spring within the tubes, okay? In this case, you may have a surface contact, but to simplify them and then you'd assume that all the contacts are somewhat uniform and symmetric. And then you can just write down the resultant contact forces, x and y direction. Same for this block, this block has a green linear guide. Also, the hold, a blue guide as well, the two guides are contacting with this red block. So first to draw the free body diagram, free the body. How many contact? Two, one with the string, one with the surface. So you have x and y components in the the string components, and the mg. That's it for the free body diagram of this block. And for this one you have to contacts with the vertical and the tilted linear guide. So there are x y components with the surface, x y components with the hold guide, plus the mg. And that's it for the free body diagram of this block, okay, so is that kind of clear? When you draw the free body diagram, just keep focus on the body, okay? And then count how many contacts does this one have? And each contact generally has an x and y component. Sometimes the direction of the forces are too obvious. Then in that case you can just draw the force in the center in that direction, plus mg. And that's it, that's how you draw the free body diagram. Before we go further, let's make a review about how we can identify internal and external forces, and action reaction pairs. Okay, sometimes in the problem you may read it as sometimes ignore the inertia effect of something, something. And what does that mean? That's why I'm going to talk about in this slide. Suppose that you have a mass A and B connected by three pulleys, okay? And the problem says ignore the inertia or the mass effect of pulley. What does that mean? Let's draw the free body diagram of this first pulley, okay? Free this pulley out of all this complicated structure. So how many contacts? One on the top, two on the bottom, left and right. So if you draw the free body diagram, there are T1, T2 and T3 and the mg, right? Now, are there any relationship between T1, T2 and T3? Are these are the same or what? Once you draw the free body diagram, we are going to learn about this more in detail later, you can actually obtain the equations of motion, F on the left-hand side, ma to the right-hand side. So all the forces on the left-hand side is T3 minus T1 and T2 here, is going to be generating the acceleration in the y direction, the positive direction, okay? And also you can draw the moment equals I alpha kind of form, which we are going to learn later. But when you have a moment arm, the radius of the pulley is R, then the torque T1 is a counterclockwise is a positive. And this one is clockwise, those torque or moment will generate the I alpha term, okay? Now if you ignore the inertia effect, that means, suppose that m equals 0, or the moment of inertia I equals to 0. And what you can have here is T3 is going to be T1 plus T2, and T1 is going to be equal to T2, right? So if you just note T1 equals T2 as T, then what you can have is T3 is 2T, and T1 and T2 are identical. So once you ignore the inertia effect, sometime the problems, or especially the pulley problems, the tensions are more simply expressed. And that's what you have to use to solve the problem. So coming back to its original, we can actually draw the free body diagram of all components. And then here you have a one contact on the top with the string, so T minus mg. This one has three contacts, but since the pulley can be ignore its inertia effect, it's a T each string, and the 2T on the top. Same for this one, there are three contacts, one T per each upward and 2T downward. And this one also had a three contact, T per each string and then 2T on the top. This box, supposed to be massless, so how many contacts? Three points, right? And then this one is a same string, so 2T upward, T upward, and then another T that actually makes the system equilibrium. So the downward tension should be 3T, right? And then also the mass B has only one contact upward with the string, and then this is a 3T, minus mg. Now, let's examine internal and external force here. Okay, suppose there is a block staying on the incline, triangular incline, which is on top of two wheel on the surface, okay? In that case the inclined surface will exert a force to the block, the normal force N. Is this normal force internal force or the external force? Try to enter this question. Yes, case-by-case, it depends on how you define the system, N could be internal or external. Let's examine a little bit further. First if you define everything as a whole system, okay? So you don't really care if the block is on the incline and the wheel, everything is just a single system. Then the only contact of the whole system with the environment is what? Yes, two contacts with the surface, right? So in this case if you draw the free body diagram, there is two x and y components of each contact, normal force and friction force of each wheel with the surface, plus mg. This m means mass of the whole system. In this case, and normal force of the incline to the block doesn't show here, this is all external forces. So in this case, normal force is internal, normal force applied by the incline to the block is internal force. However, if you separate the whole system by parts, and draw the free body diagram for each, then how many contact does this block have? Only one with the incline, so those x and y components for the external forces, plus mg. And this is a free body diagram of the block. In this case, definitely those normal force shown here is a external force. How about the incline? One contact with the block, two contacts with the wheels, so you have three contacts. So there are six forces, x and y components per each, and then the mg. How many contact does wheel have? Top and bottom, bottom with the surface, top with the incline. So there are two forces per each, so there are four forces plus mg per each wheel, okay? This is the free body diagram when you separate parts out. Now this normal force from the incline to the block is exactly same as the normal force by the block to the incline, but at the opposite direction, right? Same for the friction force. So whenever there are two objects in contact, there are action reaction forces, or pair, same magnitude in the opposite direction. So if you consider everything as a whole, those action reaction pairs are cancelled, so you only left the total external forces as shown on the left-hand side, okay? Let's do another example, a cart is pulled by the another car A, and the cart has a big wheel, shown as B. Now, there are two ways to draw the free body diagram of this cart part B. If you take everything as a whole, you don't really care. It's a wheel connected with the cart through the axle. You don't really care, I just take everything as a whole. Then the only contact of the whole system with the environment is one, the tension through the car A, and contact force on the surface, right? So two forces for each contact. This one is obvious direction so I have T tension, and x and y components at this contact, and then mg. If you separate each part, cart part and the wheel part, and suppose that those are connected by some axle, this contact point here. Then how many contacts? Just two, this cart has nothing to do with the surface. So you don't have to consider the surface friction normal forces, just keep focusing on this contact point. So there should be x and y, Rx and Ry here, and then the tension T, which connects cable to the cart, and the mg. This is the free body diagram of this cart part. How about the wheel part? One contact with the surface, one contact with the axle, so there are two forces for each x and y component. So friction and normal force and x and y components, and the mg. And this is the free body diagram of the wheel. And as I said before, those cart and the wheel, two objects in a contact at the this contact point at the center. So those Ry and Rx are same magnitude in the opposite direction, so action reaction pair. Okay, this is it about the how you draw the free body diagram. Note that regardless how complicated the whole system is, just keep focusing on the body that you want to work on. You want to get the the equations of motion, detach them, free the body. And just count, how many contacts does this one have? And each contact have, in two-dimensional, two forces exerted, okay? X and y. And this is it about the free body diagram, thank you for listening.