EQUILIBRIUM Equilibrium ◦ System of forces and resultants ◦ conditions of Equilibrium ◦ Free body diagrams ◦ Pulleys ◦ Friction ◦ Examples System of forces and resultants ◦ The resultant is simply the sum of the forces which are acting on an object. So, in this case, let's suppose we have two forces here, F1 and F2, then the resultant force is simply the vector sum of those forces. Or, more generally, the summation overall of the forces which are acting, which is shown in this equation ◦ System of forces and resultants ◦ we can form this resultant by simply adding up the components of the forces in the x and y directions. So for example the x component of the resultant force is equal to the summation of the x components of the individual forces, and the y component is equal to the summation of the y components of the forces. ◦ System of forces and resultants ◦ Let's suppose we have a body here which is acted on by three forces, F1, F2, and ◦ F3, and we'll assume that these forces do not all pass through a single point. ◦ ◦ If we want to replace that by a single force, then the single resultant is just the summation of those forces F1 plus F2 plus F3. However, if we move all those forces through a single point like that, then we can see that this is not going to be an equilibrium. Because, if I look at this point O here, each of these forces have a moment about that force So, to have those two systems be truly equivalent, I need to add a resultant moment. So, therefore, my equivalent system consists of the resulting force here, r, which is the summation of all of those forces passing through some point O here, plus a resultant moment M0, which is the sum of the moments of all of those individual forces about the point. So, in other words, the resultant force is just the summation of the forces which are acting, in this case, F1 plus F2 plus F3, and the resultant moment is the sum of the moments of the individual forces about that point. Now we can go further than that and replace this force system by a single force acting at some point. ◦ So let's suppose that my single force here is R. And if I want this to be a single force here, then I have to, for this system to be in equilibrium, the moment of my resultant force about the point O, in other words, R times d, must be equal to the summation of the individual moments. In other words, the distance here of my resu, single resultant force is equal to the moment, the result in moment divided by the resultant force. So in this way I can replace a force, a system of forces by a single force with no moment. However, I have to calculate where that single force the line of action of that single force. ◦ The conditions for equilibrium are, generally speaking, the resultant force or the summation of the forces acting is 0, and the summation of the individual moments about some point is also equal to 0. Those are our basic conditions for a body to be in equilibrium. ◦ There are some special cases of that, as shown in this diagram on the next page. ◦ The first one, collinear, means that the forces all have the same line of action. So they all pass through one line as shown here. And, in this case, we have only one equation of equilibrium that the sum of the forces along the line of action of those forces, which I'll call the x axis, is equal to 0. ◦ ◦ ◦ The second special case is the case where we have a system of forces which are concurrent in other words, they all pass through a single point. In that case, in two dimensions at least, we have only two equilibrium conditions, that the sum of the forces in the x direction and the sum of the forces in the y direction is equal to 0. We don't have a moment equation, because all of the forces pass through this point O, therefore they don't have any moments about that point. ◦ The third special case is a parallel force system, where all of the forces are parallel to each other. And here again, we have only two equations of equilibrium. We have that the equations of, of the sum of the forces in the direction parallel to the forces, which I'll call the x direction here, is equal to 0. But also the sum of the moments of all of these forces about that point must also be 0. So we have two equations. ◦ The most general case, where we have a system of forces and moments, we have three equations, again in two dimensions. We have the sum of the forces in the x direction equals 0, the sum of the forces in the y direction equals 0, and also the sum of the moments about some point is also equal to 0. So, generally speaking, in two dimensions we have three independent in eq, equilibrium equations, which could, for example, be sum Fx is equal to 0,sum Fy is equal to 0, and sum of the moments about some point is equal to 0. However, they don't need to be those three. Equally, we could have sum Fx is equal to 0,sum of moments around some point A is equal to 0, or sum of some moments about B, some point B is equal to 0. So in either case we have only three equilibrium equations, not more. ◦ If we have three unknowns, then we have three equations, therefore we can solve for those three unknowns. And we call that kind of a system statically determinate, which is the kind of system we'll be dealing with in this segment on statics. ◦ If there are more than three unknowns then we can't solve it from statics alone, and we call that kind of a system statically indeterminate. And to solve that kind of system we need some other information, for example, relationship between the deformations and the forces. And that topic we'll look at later on in mechanics and materials. ◦
