[MUSIC] Hello, today we're going to talk about the effect of a System of Forces on a body. And under what conditions to different system of forces acting on the body, are equivalent in the sense that they will cause the same motion of the body assuming the same initial conditions. We will consider here rigid bodies. The body is called rigid, if the distance between any two points, remains unchanged no matter what forces acting on the body. For example, if we consider this cup, this is clearly a deform of a body. It's not there is a body because when we apply forces, this, the distance between this two points for example, changes. So, assuming that we have a rigid body. This body could be acted upon by a system of forces. So that Fi. Fn And, by a system of couples with moments M1, M2, let's say Mj, let's say Mm. These forces and these couples, they will cause this body to move, to accelerate and rotate and spin with some angular acceleration. And the question is, under what conditions to different system of forces have the same effect in the kinematics of this body? So, we will consider two system of forces acting supposedly on the same body. I will show only one of these forces for example let's say, we consider the forces Fi and the couples, Mj, acting on his body. Where Fi i varies let's say from one, let's say we have N such forces, and where the moments Mj, the couples Mj, vary from one to M. So we have M sets couples. At the same time, we consider this body acted upon by a different system of forces. Let's call this F prime i. And couples let's say, Mj prime. These are all vectors of course, where in this case Fi prime, Varies from one to let's say N prime. So, we have N prime such forces. They could be the same number or different number of forces. And moments, let's say Mj prime, where j goes from one to M prime. Again we could have the same or different number of couples acting on the body. So the question is, when do we call, when do we say that these two systems of forces are equivalent? We say that these systems of forces are equivalent is the following conditions are met. Firstly, the sum of all the forces, for this body, should be equal where I goes from one to N, is equal to the sum of all forces F prime. Where I goes from one to N prime. That means, the sum of all the forces in this two cases are the same, because the sum of all the forces will control the translational motion of this body. As a matter of fact, the sum of all the forces will be equal to the mass of the body times the acceleration of it's center of mass. So if you want to have the same acceleration of the center of mass, we must make sure, the sum of the two forces, are equal. Secondly, in order to guarantee also the rotation equivalence of these two system of forces. If we pick up a point A, here. What should be true, is the total rotational effect of this system of forces and couples, for this system and for these systems are equal. So, what we will require Is that the sum of the moment, about point A is equal to the sum of the moments about point A for the prime system. Okay? This, the rotational effect about point A, is the result of the rotational effect of the forces and the couples. So this guy, can be written as the sum of all the rotational effects of the forces Fi, which can be defined as ri, the sum of all the ri's cross Fi, plus this is for all i's, plus the rotational effect of all the couples which is the sum of all the Mj's, where j goes from one to M and i goes from one to N. At the same time, the rotation effect about point A for this system, would be the sum the rotational effects of all the forces Fi prime. So we would have to define vectors ri prime. Where i, goes from one to N prime. Plus the rotational effect due to all these couples, which is the sum of all the moments Mj prime. Where j prime, where j goes for one to M prime. So, under this condition, we guarantee that the total rotation effect about this point is equal for these two system of forces. So this obviously, are necessary conditions to guarantee equivalents, we must have translational equivalents and rotational equivalents about point A. Of course, as you would expect, it is not sufficient, it is what we require for equivalence of these two systems, is that the moment the rotation effect about any other point B, any other point B here, will be also equivalent. That means, we would like also the sum of the moments about point B for the original system and for the prime system to be also equal. So the equivalence should be not only with respect to one particular point, but with respect to any point. So basically, the necessary conditions for equivalence of these two systems are infinite, because for every point we have to guarantee equivalence. What we are going to show today, is that if we guarantee equivalence of forces, and equivalence of moments with respect to a particular point, then we automatically guarantee equivalence with respect to any other point. And this is a very important result. That means, although the equivalence about any point is necessary condition, this conditions are sufficient to guarantee all these necessary conditions. So, the necessarily sufficient conditions for equilibrium, reduce to a total of two vector equation, one vector equation another vector equation, which is three equations in terms of components when we're talking about the three dimensional problem and another three equations for this one for the moment, equivalence. Because a vector equation, means that we have to guarantee the x, y and c components of the two sides are equal. In the case where we have a two dimensional problem, the number of necessary sufficient conditions reduces because in that case we do not have any z components here. So, this is true for a 3D problem. But for the 2D problem, the equivalence of forces reduces to two equations because if we have a two dimensional problem, we only have x and y components, we don't have any z components to start with, and the moment equation, since all the moments are in the z direction, we have only one equation, to guarantee the equivalence of the moments. So, we're going to prove next, that once we guarantee these conditions, that automatically the moment about any other point, these moments are automatically equal. So, what we're going to prove now is that given these conditions, that means equivalent of forces and equivalent of moments about one point, we will prove that we have equivalent of moments about an arbitrary other point B. So, let's express the sum of the moments about B. The sum of the moment about B, is the total rotational effect about point B, is the rotational effect due to the force of i. For that I would have to take position vectors, let's call it ri tilde, okay? So, would have the sum of let's say, Ri tilde plus Fi Plus the rotational effect due to the couples. Sum of the moments, Mj, j goes from one to M, i goes from one to N. However, the position vector ri tilde, can be expressed as the sum of, BA, the vector of BA plus ri Cross Fi. And using the distributed property, this can be written sum of BA is Fi. Which is BA cross with the sum of Fi's. [BLANK AUDIO] Plus the sum of ri's cross Fi's. From i equal to one through N. Plus the sum of all the Mj's. Now we notice that these two last terms, are exactly what we have here. So, this tells us that the moment about point B, is BA cross the sum of all the forces Fi, [BLANK AUDIO] plus the sum of the moments about point A. We could do exactly the same for the right system and what we would find is that the sum of the moments for the prime system about point B, would be for the same reasons as BA cross with the sum of the forces Fi prime i1 to N prime, plus the sum of the moments of the prime system about A. Now since we already guaranteed this, it means that this term and these term are already equal. And since the sum of the forces, Fi and the sum of force Fi prime are equal, this term and this term are also equal. Therefore, it follows automatically that the sum of the moments upon point B for the original and the prime systems, remain equal. So we proved, that the necessary and sufficient conditions for equivalence of these two systems of forces is equivalence of the sum of the forces, and the equivalence of the sum of the moments about an arbitrary point A. These conditions guarantee also equivalence of the rotational effect of the system of forces about any other point, and therefore these two systems will behave kinematically, exactly the same assuming we have the same initial conditions.
