Hello. In this video, we will see how to explain the equilibrium of two persons who are practicing tug-of-war. We will see how the weight of the persons intervenes, how we can determine the internal force in the cable, and what are the effects of the ground on the persons who are practicing tug-of-war. A very simple exercise, it is an exercise of tug-of-war in which both persons are exactly identical. For this case, we have a person on the left, with a weight of 800 Newtons, a person on the right, with a weight of 800 Newtons. Then we are going to have an internal force in the rope, and an internal force under the feet of the person, which we will be able to reveal by means of a free-body. Actually, we want to take an interest in three free-bodies, the free-body of the man on the left, cutting just under his feet, and cutting the rope. The free-body of the man on the right, which is exactly the same. And finally, a little free-body with a segment of cable on the middle, whose you will see that it is useful for us anyway. So, first, we have the weight, the 800 Newtons of each of these two persons. We have the effect of the rope on the man on the left, on the intermediary segment and on the man on the right. So, a priori, we are going to say that there is a tension T one, and a tension T two. But if we have a look on the equilibrium of the free-body in the middle, we will see that there are, acting on this free-body, only two forces, the force T one which pulls towards the left, the force T two which pulls towards the right. Thus, necessarily, T one must be equal to T two, and then, is equal to T, which we will use in the following. So, we needed this free-body anyway. If we extend the line of action of the force T on the left and on the right, it will cut the line of action of the force of the weight. And, to have the equilibrium, it will be necessary that the force under the feet should be converging with these two other forces. Then, that these three forces are converging. And we thus get, that the force, under the feet of the person on the left, S one, passes by this point. And likewise, the force under the feet of the person on the right, S two. Let's draw the acting forces, for the free-body on the left in the Cremona diagram. So, here, we have the weight of 800 Newtons. We are going to have the force T which acts on the horizontal axis. We are not totally sure until where. Then, we know, on the other hand, the direction of the force S. I make the transition, a little bit quickly. Excuse-me. So, we get the force S one, and the force T which express the equilibrium of the free-body on the right. Thus, this person is in equilibrium, and we can read graphically the value of the internal force in the cable, which is equal to T. Let's now take a look at the equilibrium of the person on the right. To do that, we are going to reuse the segment T that we have already drawn. In this way, we are not going to introduce any mistakes. So, we are going to introduce the new force of 800 Newtons. We are going to reuse the segment T, in the other direction, and we are going to close the polygon of forces expressing the equilibrium of the person on the right, by the force under the feet, S two. This construction, in which we combine two polygons of forces, together, with one or several shared forces, is called a Cremona diagram. This Cremona diagram will be a construction that we will use very often, thereafter. So, it is very important to understand well how we construct it, and to see well the meaning of the various components of this diagram. Let's now look at a tug-of-war a little bit more realistic, between two persons. The person of 800 Newtons, that we know well, and another person evidently lighter. It is quite clear that the lighter person was not able to stay in equilibrium, when the heavier person, on the left, began to lean back. But let's try to see why, in the Cremona diagram. We have, here, on the left, a person of 800 Newtons, and on the right, a person of indeterminate weight, but clearly lighter. We draw the line of action of these two forces, since we will need them after. And we draw the line of action of the internal force in the cable. This construction enabling us to obtain the direction of the internal force under the feet. We can see that this person, in this case, did not really lean on the heels, on tip-toe. So, here, we have S one. Then, we can notice that the second person is more leaned back with an internal force S two, under her feet. Let's now have a look at the Cremona diagram, with the force of 800 Newtons, the direction of the internal force T, the tension in the cable. And, we copy the inclination of the person. It gives us the complete solving of the equilibrium of the person on the left. Then, with the internal force T in the cable, and the internal force S one under the feet. Let's look at what is about the person on the right. Using the construction of the Cremona diagram, we already know the internal force T, and we directly obtain the internal force S two, under the feet of the person, and, we deduce from that what must be the weight of this person. We do not need to know it, but we could determine it graphically. What we can notice that the inclination of the person on the left, alpha one, is lower than the inclination alpha two. We can see it even more clearly here. We have alpha one, here, alpha two. Just a this moment, we still have the equilibrium, but you have seen in the video that just after, the person on the right loses her equilibrium, while the person on the left leans down a little bit more. Why is it possible ? Well, precisely because, if we supposed that both persons had the same friction capacity, then, the person on the left could increase more his inclination without sliding. Actually, if we take a closer look on the feet, we notice that the person on the left had very favorable shoes. So, in all likelihood, the alpha max of one, is greater than the alpha max of two. Thus, the person on the left would have been able to lean back more than the person on the right and would have won. But on the other hand, if we come back to the previous diagram, what we can see is that the person on the right, because she is lighter, must, anyway, in all cases, lean back more than the person on the left to offset the tensile internal force T. I should have put the second arrow on T for the Cremona diagram to be complete. So, in these conditions, the person on the right has simply no chance to win. Even if, actually, both friction coefficients would have been equal, she would not have any chances. The only configuration, in which the person on the right could have a chance, that is if the heaviest person wore, for example, roller skates. In this way, the person on the right had a chance to win this game. We can ask ourselves the question, when we see that, to know if tug-of-war is really a sport, since actually, what counts is a friction coefficient. We can imagine that all the participants will have good shoes and a good friction coefficient, and on the other hand, it is the weight of the persons. So, obviously, the heavier team is going to win. In conclusion, we have seen in this video on tug-of-war, how to isolate the free-bodies which enable us to determine the individual equilibrium of each person. We have been able to construct the polygon of forces which expresses the equilibrium of each person, and combining these two polygons together, we have created a new construction which is called the Cremona diagram, which will be very useful for the rest of this course. We have also seen the dominating influence of the weight, and of the friction coefficient in tugs-of-war.
