 In today's session I am going to deliver lecture on Equilibrium, basically we focus on Spherical Cylinder, Myself, Bipin Patil, Assistant Professor, Department of Civil Engineering, Valchan Institute of Technology, Solarpoor. These are the learning outcomes for today's session. So at the end of this particular session, learners or students will be able to first one, explain the concept of equilibrium, what do you mean by equilibrium and second, how to calculate the forces present in connected bodies. Let's we discuss the concept of equilibrium. First on this particular figure, body showing result on an equilibrium, two forces are present on this particular body, the force system is present here, first one force P, second one Q, how to calculate the result on the resultant of this particular force system. So we just use log parallel ground and according to that we just try to identify the resultant. So there are different methods, either you calculate analytically, graphically and mathematically, that is nothing but you just use the formula and you just try to identify this resultant. So equilibrium is applicable to those system of forces whose resultant action is 0, that is E is equal to R. So R is nothing but your resultant, E is nothing but your equilibrium, E is equal to R, means here by considering this particular P and Q forces, you just try to identify the resultant force that is R. Now the magnitude of this particular resultant, you just calculate it, the same direction means R is present here, so exactly in opposite direction, equilibrium is there, that is nothing but the equilibrium force is there. So line of action is same, same magnitude but the direction is different. So basically few conditions are present, we called as analytical conditions of equilibrium. So the same figure is present here, number of forces acting on this particular body, how to identify this particular unknown forces, for that we are considering here two laws, first one is force law, algebraic sum of forces acting on a body, it may be P, it may be Q or so on. So just consider the sum of all this particular forces and equate it to 0, this law is called as force law. So here we are considering summation H is equal to 0, that is all horizontal forces are 0, summation V is equal to 0, that is all vertical forces are 0. Second law, movement law, means you just consider any one movement center, it may be concurrent forces, non-concurrent forces, the algebraic sum of movement of forces about any point, means where you are considering movement center, so you just equate it to 0, that is nothing but movement law of equilibrium, that is summation M is equal to 0. Few steps we require to identify these particular forces, unknown forces, when body is in contact with other body, first step identify the contact surfaces, you just observe this particular figure in your problem statement and identify the contact surfaces, step 2, draw the free body diagram by removing contact surfaces, means you just identify the contact surfaces and remove that particular contact surfaces and show the reactions in opposite direction. Step 3, apply the equations, it may be lambs theorem and static equilibrium equations. So these two particular conditions you just apply to identify the unknown values and last step to find out the unknown forces. Let us we discuss one problem, a sphere of weight 50 kilotons, weight of this particular sphere they are given and the radius of this particular sphere they are mentioned 10 centimeter, arrest in a channel shown in figure, find the reactions at a point of contact. So two contacts are present here, one is here, number one, second one is number 2 here and the inclination they are mentioned with respect to x axis 60 degree at one side and 30 degree at another one side. Now my dear friend these are nothing but your two contact surfaces, remove that contact surfaces and show the reactions in upward direction. So here we are going to apply two conditions, one is lambs theorem, second one is static equilibrium equation. So see here this particular picture is present here, this is the free body diagram of this particular picture. So you just observe here initially O point that is your center, weight it acts in downward direction so I mentioned here, R1 first reaction at first contact surface, R2 second reaction at second contact surface. Now see here lambs theorem we are going to apply either all these forces acts the center of this particular point towards the point or away from the point. So according to that we just prepare a geometry and this angle is similar to this particular inclination with respect to x axis. So R2 30 degree with respect to y axis and R1 60 degree with respect to x axis. Now here we are considering with respect to y axis. So now you just forward this particular force in this direction, R2 you just forward it in forward direction and you just identify this particular angles. So x and y coordinate you just consider here x axis y axis according to that you just calculate the remaining angles. So see here how to calculate this particular 120 or 1 force where I just forwarded in this particular direction R2 force you just forward in this particular direction. Now how to calculate this 120 degree? See here this particular angle is 60 degree now extend this particular line here. So with respect to y axis this angle is 60 degree. So with respect to x axis 30 degree is there and from here this is 90 degree. So 90 plus this is 30 that is nothing but 120. Similarly here 30 you just extend here, 60 you just extend here. So this angle is 90 degree. So already we have calculated here. Now remaining one 360 degree minus addition of these two it gives you remaining value. So this is the correct free body diagram by considering this particular geometry. Now apply Lamis theorem. In Lamis theorem you just consider this force and the angle between remaining these two forces sign up angle between remaining two forces. So opposite this particular direction you just consider it. So first one R1 and sign of this particular angle 150 R2 sign 120 is there and weight W1 sign 90 is present here. Now consider the weight of this particular this geometry that I mentioned here 50. So we just put the value R1 sign 150 W1 sign 90. So R1 and we just consider here W1. So W1 that I mentioned 50 put the value of W1 here and calculate the value of R1. So R1 it comes 25 kilotron that is your equation number 1. Now consider second W1 upon sign 90 equate it to R2 sign 120 put the value of W1 calculate the value of R2 that is 43.30 kilotron. So let us check this particular values do the calculations now my dear friends we just pause this particular video and try to give the answer of this particular question. So this is the answer of this particular question to prepare this particular session I refer this particular references thank you.