 I am Swati Gharge, Assistant Professor, Department of Civil Engineering from Vulture Institute of Technology, Solapur. Topic for today's session is Equilibrium of Concurrent Force System. At the end of this session, learner will be able to analyze Concurrent Force System. Let us introduce first what is Force System. Force System is a collection of forces acting at a specific location. Depending upon the position of forces, the force system is classified in several types. Force System is basically classified in two types, Coplanar Force System and Non-Coplanar Force System. Coplanar Force System means all the forces are acting in one plane and in non-coplanar forces they are not acting in one plane. Another force system again classified in two following types that is Concurrent Force System and the Non-Concurrent Force System. We will see now Concurrent Force System. Here there is one body on that body, several forces are acting A, F1, F2, F3 and F4. All they are acting at point O means all the forces are meeting at point O. Their meeting point is same so the force system is called as the Concurrent Force System. Now you think and answer the question, Force System shown in figure is Concurrent Force System or Non-Concurrent Force System? Now here is the answer. Your force system is given as a rectangle having four corners A, B, C and D. At A there is a one force is acting, at B also there is a one force is acting. At point C also there is a one force acting and at point D there is a one force acting. So initially it is looking like a non-concurrent force system but after extending the line of action of all the forces, we come to know that all the forces are meeting at one point and that is at the center of rectangle their meeting so it is a Concurrent Force System. The equilibrium equation used for the Concurrent Force System are summation of Fx is equal to 0 and summation of Fy is equal to 0. Now we will solve that numerical. Here two identical smooth sphere of radius 100 mm and weight 100 Newton rest in horizontal channel having vertical wall. Find the reaction at point of contact A, B, C and D as shown in figure. Now we will draw first the free body diagram of this two sphere. This is of self weight of cylinder 1 is given 100 Newton. It is acting vertically downward because it is a self weight of cylinder 1. Again self weight of cylinder 2 is also 100 Newton because it is given their identical cylinder so their weight and their size are same and their self weight is acting at the center and vertically downward. So two forces we have shown in a free body diagram. So at point A it is supported so remove that support and instead of support show support reaction. So as the support is vertical see you can see here the support is vertical so after replacement of this vertical support will show the support reaction horizontal and we will name it as a support reaction at A. At point C here look at point C here this is a horizontal surface of channel. So after replacement of this horizontal surface support reaction will be perpendicular to the surface so it is vertical and say it is RC. Now look at point D. Point D is the contact point between cylinder 2 and the vertical wall. So after replacement of this vertical wall that support reaction we will show horizontal because reaction is always perpendicular to the support and we will name it as a RD. So this is a free body diagram of these two cylinders cylinder 1 and cylinder 2. For individual cylinder for cylinder 1 we make it separate cylinder 1 and cylinder 2. As they are in contact with each other so there is a mutual reaction between these two cylinders. So here I have shown the mutual reaction between these two cylinders and we have named that point as a B so reaction is RB. So this is a complete free body diagram of cylinder 1 and 2. This is a given figure in that RB is the line joining the two centers of cylinder and here there is a triangle formation because we have to find out the angle of this force RB. So this is a triangle with the help of the triangle we can determine the angle of this RB with horizontal or vertical. Here we will determine with horizontal. So we can find out the RB. RB is look at the figure it is very easy to understand the RB is nothing but the addition of two radius. Radius of cylinder 1 and the radius of cylinder 2. It is total RB. And look at the base of this triangle. Base of the triangle is we know that the distance between two vertical wall is 360 mm and from that 360 mm if we subtract the radius of cylinder 1 so here the dotted line I have shown that is the radius of cylinder 1. And again if we subtract the radius of cylinder 2 the remaining distance will give you the base of the triangle. So we have the hypotenuse and base of the triangle with the help of this two dimension we can easily determine the angle of that inclined force RB. So here I have shown in this figure look at this figure and now from the trigonometric rule determine theta. Theta will come 36.87. So RB is making 36.87 with horizontal so here also I have shown 36.87 with horizontal and at this end also you can show as their alternate angle so they must be same. Now consider the cylinder 1 only. So cylinder 1 having three forces one is a self weight second is the support reaction at point A and third is the mutual reaction between two cylinder that is RB. So three forces are there so it is very easy to analyze that force system by lambs theorem. So for using a lambs theorem we must know all the angles between these three forces. So here the 100 Newton is vertical RA is horizontal so no doubt that angle will be 90 degree. And we have determined theta so with the help of that theta plus 90 degree will come 126.87 and this two angle you subtract from 360 degrees so that 143.13 will come. Now we can apply lambs theorem very easily. So by applying lambs theorem what is lambs theorem that is it is the constant ratio of the force and the opposite angle of sign. So here 100 Newton force it is the first force that is known to us and opposite of 100 Newton angle is 143.13. So ratio is 100 divided by sine 143.13 that must be equal to another force that is RA and opposite angle is 126.87. So RA divided by sine 126.87. This must be equal to third force that is RB and opposite angle of RB is 90. So RB divided by sine 90 all these three ratio must be same because the force system is in equilibrium. So after solving this equation we will get RA is equal to 133.33 Newton and RB is equal to 166.66 Newton. Now we will analyze the cylinder 2. Look at the free body diagram there are total four forces one is self weight that is 100 Newton reaction at point C reaction at point D and the mutual reaction between two cylinder that is RB. So here lambs theorem is not applicable because forces are more than three. So we will solve it by using equilibrium equation. So we have two equilibrium equations summation fx is equal to 0 and summation fy is equal to 0. We will use the first equation that is summation fx is equal to 0. So here RD is in x direction and the x component of that RB will be in x direction. So let us determine first x component of RB. So x component of RB is RB into cos 36.87 and we will determine the vertical component also that is RB into sine 36.87. So here we have resolved RB in x and y direction with the help of this theta that is 36.87. Now see here two forces are there in x direction RD and RB cos 36.87 as they are in equilibrium so that two forces must be same. So RB cos 36.87 is equal to RD the values of RB in the equation and get the value of RD it will come 133.32 Newton. Now we will use another equation that is summation of vertical forces is 0. In this free body diagram total three forces are in y direction one force is 100 Newton second force is RC and third force is RB sin 36.87. So as it is in equilibrium summation of vertically upward forces must be equal to summation of vertically downward forces. So RB sin 36.86 plus 100 must be equal to RC. So put the value of RB in this equation and find out the RC that is it will come 200 Newton. These are my references for this video. Thank you very much.