hello in this lecture we will learn about couples and also about Force couple systems so what is a couple well by definition a couple of forces is the moment produced by two equal opposite and non-colinear forces or in other words parallel forces so it is a vector it is a vector that has the meaning of a moment so if we assume a flat rigid body and two forces F and plus F acting in the the plane of this body these forces have equal magnitude opposite directions and they are parallel and they're also located at distance D away from each other we can consider a point O and determine the moment of this force is about at Point O by also knowing the distance from o to F being say a so in order to calculate the moment of the two forces we will sum the moment produced by each Force so f * a + d - f * a and you can see that a will cancel out and the resulting magnitude of the moment is f * D the moment Vector of course will be perpendicular to both forces and in this case it will rotate counterclockwise so it is a positive moment okay so what you can see from the above relationship is that distance a the distance to point O has cancelled out and the moment is only dependent on the distance between between the two forces and is independent on the point about which we calculated the moment of a couple so that makes the moment a free Vector meaning that it can act anywhere and its rotational effect on the rigid body will be the same now note and remember that forces were sliding vectors so you could always always you would always be allow but only to slide the vectors along their lines of action you could not move them away from the their lines of action that made the forces sliding vectors whereas moments can be moved away from their lines of actions hence making them free vectors A few words now about the sign convention for couples which which is very similar to the sign convention for regular moment so in a case like this one where the two forces creating the couple tend to rotate counterclockwise using the right hand rule we will obtain a direction of the moment Vector out of the page and that would create a positive moment because it will be in the positive direction of the Z axis now for these forces couple of forces that will rotate clockwise they will generate a negative moment Vector which will be directed into the page so in the negative direction of the Z axis the other thing that we need to to look at is if we have couple of forces acting on Parallel planes so in the three-dimensional space we can have two forces acting on a plane like for example this one and we can also have the same forces acting on a separate a different parallel plane parallel to the original plane and given that the magnitude of the forces and the direction of the forces is the same then they will create a moment Vector each one will be at Point O Prime and one will be at Point O and obviously the moment being a free Vector then it will be it will have the same effect on either plane so in other words the moment is the same regardless on where the couple of forces act on which of the parallel planes the couple of the forces acts now remember free vectors are also sliding vectors so you can not only move them away from their line of actions but you can also move them along their slide of their line of action however sliding vectors are not free vectors sliding vectors you can only move them along Ong their lines of action and not away from their lines of action however you can actually replace a given Force by an equal and parallel force that is away from the line of action of the original Force but that would obviously violate the principle of sliding vectors however you can do that by compensating for the move by adding a couple which will account for the change of the line of action of the force so how can this be done well let's see a quick illustration in which on a rigid body there is a force F acting at point B as shown and in the end we would like to move the force at Point a and the end system has to be equivalent to the original system so let's do this one step at a time and I've drawn again the body with the force now shifted to point a these two systems have to be equivalent so the resultant force of the systems must be the same that's condition number one which it is and also the moment taken by a point let's say point a on the first system must be equal to the moment of the all forces on the second system about the same point a so in this case this is also satisfied because what we've done we've added a negative Force F at Point a which will compensate for the positive force F and we've also added back the original Force f at point B so these systems are indeed equivalent now in the second step we will have of course the force F at a but we're going to replace the negative Force F and force F in the second diagram with a moment because these two forces as you can see here are equal and opposite and of course parallel so they create a couple the magnitude of the couple can be calculated by multiplying the force times the distance between the forces the moment arm which is D so now the last system is also equivalent with the original system because of course the resultant of all forces is f and the sum of the moments about a will be equal to the moment m so in the end F and M represent a force couple system you can always again move the force from its line of action but you have to add a moment so in effect you create a force couple system so thank you very much for watching this lecture and I'll see you at the next one