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1 Lecture 2: Equation of Equilibrium A particle is in equilibrium if it is stationary or it moves uniformly relative to an inertial frame of reference. A body is in equilibrium if all the particles that may be considered to comprise the body are in equilibrium. One can study the equilibrium of a part of the body by isolating the part for analysis. Such a body is called a free body. We make a free body diagram and show all the forces from the surrounding that act on the body. Such a diagram is called a free body diagram. For example, consider a ladder resting against a smooth wall and floor. The free body diagram of ladder is shown in the right. Three forces are acting on the ladder. Gravitational pull of the earth (weight), W of the ladder, reaction of the floor R 2 and reaction of the wall R 1. When a body is in equilibrium, the resultant of all forces acting on it is zero. Thus that resultant force R and the resultant couple M R are both zero, and we have the equilibrium equations These requirements are necessary and sufficient conditions....(2.1) Let us understand equation (2.1) for different type of force systems. 1. Collinear forces: In this system, line of action of all the forces act along the same line. For example, consider a rope being pulled by two players. Suppose one player is pulling with a force F 1 and the other is pulling with a force F 2. Then the resultant force, in the direction of pull of first player is (F 1 F 2 ). By the first condition of (2.1), it should be zero. Hence, for equilibrium F 1 =F 2. With this, the second condition will also be fully satisfied, because moments of the forces about any point in the space will be of equal and opposite magnitudes. Hence, the resultant moment will be zero. contents/iit %20Guwahati/engg_mechanics/chap1/print/lec1/lec2.htm 1/12
2 2. Coplanar parallel force: In this system, all forces are parallel to each other and lie in a single plane. Consider a See Saw. Two children are sitting in the See Saw and it is in equilibrium. If the weights of both the boys are W 1 and W 2 and they are sitting at a distance of d 1 and d 2 from the fulcrum, then for equilibrium, the fulcrum reaction should be...(2.2) Taking moment about fulcrum, W 1 d 1 =W 2 d 2...(2.3) SEESAW 3. Coplanar concurrent point: In this system, line of action of all forces pass through a single point and forces lie in the same plane. An example is the weight suspended by two inclined strings. Here, all the three forces acting on a particle pass through point O. The moment of the forces about the point O is zero. Hence, second condition of (2.1) is already satisfied and only first condition needs to be checked. 4. Coplanar non concurrent forces: All forces do not meet at a point, but lie in a single plane. An example is a ladder resting against wall when a person stands on a rung, which is not at its center of gravity. In this case, for equilibrium, both the conditions of ladder need to be checked. contents/iit %20Guwahati/engg_mechanics/chap1/print/lec1/lec2.htm 2/12
3 5. Non coplanar parallel forces: In this case, all the forces are parallel to each other, but not in the same plane. An example is forces on the table, when a book is kept on it. Here, both the conditions have to be checked. 6. Non coplanar concurrent forces: In this system, all forces do not lie in the same plane, but their line of action passes through a single point. For example, if a disc of weight W is suspended by means of three strings, line of action of all the forces pass through point O. The forces do not lie in a plane. The second condition of equation(2.1) is automatically satisfied. For equilibrium the first condition is to be checked. 7.All forces do not lie in the same plane and their lines of action do not pass through a single point.this is the most general case and both the conditions of eqn.(2.1) need to be checked. For Equilibrium more visit of coplanar our Website concurrent forces: contents/iit %20Guwahati/engg_mechanics/chap1/print/lec1/lec2.htm 3/12
4 If all the forces in a system lie in a single plane and pass through a single point, then the system constitutes a coplanar concurrent force system. The resultant R of a system of forces is zero if and only if it's resolved components in two orthogonal directions x and y are zero. Moment about the comm contents/iit %20Guwahati/engg_mechanics/chap1/print/lec1/lec2.htm 4/12
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