CHAPTER III. STATICS OF CONSTRAINED BODIES, ETC. 72. THE Conditions of equilibrium which we have proved in the last Chapter apply to any rigid bodies whatsoever. If however the body considered be a constrained one, i.e. one that is not free to move in every way, as for instance one that can only turn about a fixed axis, we can obtain conditions of equilibrium which do not involve the forces of constraint. 73. Prop. If a rigid body under the action of a system of coplanar forces, have one point in the plane of the forces fixed, it is a necessary and sufficient condition of equilibrium that the algebraical sum of the moments about the fixed point of the forces, excluding the force of constraint, be zero. For the force of constraint acts through the fixed point A, and therefore when there is equilibrium, the resultant of the remaining forces must act through A. But the algebraical sum of the moments of these remaining forces about any point is equal to the moment of their resultant, and therefore that about A vanishes. The condition is therefore a necessary one. It is also sufficient. For if it hold, it can be shewn as in Art. 52 that A being a fixed point, the body is in equilibrium. 50. A set of equal frictionless cylinders, tied together by a fine string in a bundle whose cross section is an equilateral triangle, lies on a horizontal plane. Prove that if W be the total weight of the bundle and n the number of cylinders in a side of the triangle, the tension of the 1+ W string cannot be less than (1 + 1) 4/3 -1 or W 4/3 (1-1), according as n is an even or an odd number, and that these values will occur when there are no pressures between the cylinders in any horizontal row above the lowest. 51. A quadrilateral ABCD has the sides DA, AB, BC equal and the angles DAB, ABC right angles, but AB and CD are not in the same plane. If forces acting along the four sides can be reduced to a couple, its axis will make with AB an angle 52. Forces act along the edges BC, CA, AB, OA, OB, OC of a finite tetrahedron, represented in magnitude by XBC, μÑA, vÁB, X'OA, μ'OB, v'OC respectively. Prove that they will be equivalent to a couple, if X' + μ ¬v=μ' + v − λ = v' + λ − μ=0. 53. Prove that the axis of the resultant of two given wrenches (R1H1) and (RH), the axes of which are inclined to each other at an angle 0, intersects the shortest distance (2c) between their axes at a point the distance of which from the middle point is (R12 - R2) c + (HR2-HR) sin CHAPTER III. STATICS OF CONSTRAINED BODIES, ETC. 72. THE conditions of equilibrium which we have proved in the last Chapter apply to any rigid bodies whatsoever. If however the body considered be a constrained one, i.e. one that is not free to move in every way, as for instance one that can only turn about a fixed axis, we can obtain conditions of equilibrium which do not involve the forces of constraint. 73. Prop. If a rigid body under the action of a system of coplanar forces, have one point in the plane of the forces fixed, it is a necessary and sufficient condition of equilibrium that the algebraical sum of the moments about the fixed point of the forces, excluding the force of constraint, be zero. For the force of constraint acts through the fixed point A, and therefore when there is equilibrium, the resultant of the remaining forces must act through A. But the algebraical sum of the moments of these remaining forces about any point is equal to the moment of their resultant, and therefore that about A vanishes. The condition is therefore a necessary one. It is also sufficient. For if it hold, it can be shewn as in Art. 52 that A being a fixed point, the body is in equilibrium. Ex. 1. A uniform rod which is 12 feet long and which weighs 17 lbs. can turn freely about a point in it, and the rod is in equilibrium when a weight of 7 lbs. is hung at one end. How far from that end is the point about which it can turn? Ans. 4 ft. 3 in. Ex. 2. ABCD is a square: a force of 1 lb. acts from A to B, one of 4 lbs. from B to C, and one of 15 lbs. from D to C: if the centre of the square is fixed, find the force which, acting along DA, will maintain equilibrium. Ans. 10 lbs. Ex. 3. ABCD is a square, of which the point A is fixed: a force of 2 lbs. acts along AB, one of 6 lbs. along AD, one of 10 lbs. along BD, and one of 3 lbs. along BC, find the force along DC which will maintain equilibrium. Ans. (5√2+3) lbs. Ex. 4. A lever ABC, with a fulcrum B, one-third of its length from A, is divided into equal parts in D, E, and F. At C, D, and F, forces of 12 lbs., 8 lbs., and 6 lbs. respectively act vertically downwards, and at E a force of 16 lbs. acts vertically upwards. What force applied to A will cause equilibrium? Ans. 21 lbs. Ex. 5. A weightless lamina in the shape of a regular hexagon ABCDEF, is suspended from the middle point of AB: shew that it will be in equilibrium with the side AB horizontal, if weights of 3 lbs., 7 lbs.,. 3 lbs. and 5 lbs. are hung at C, D, E, and F respectively. 74. Prop. If two points of a rigid body be fixed, so that it can only turn about the line joining them, it is a necessary and sufficient condition of equilibrium that the algebraical sum of the moments of the forces, excluding those of constraint, about the fixed line, be zero. If there is equilibrium, the algebraical sum of the moments of all the forces about any line is zero, and the moment of the force of constraint at each of the fixed points about the line joining them is zero: therefore the sum of the moments of the remaining forces, excluding those of constraint, about this line, is zero. It is therefore a necessary condition. It can be proved as in Art. 54 that when the algebraical sum of the moments about any line is zero, there is equilibrium provided two points in the line be fixed. The condition is therefore sufficient. 75*. Prop. If one point of a rigid body be fixed, the necessary and sufficient conditions of equilibrium are, that the algebraical sum of the moments of the forces about each of three lines through the fixed point, but not in the same plane, be zero. It can be shewn, as in the last proposition, that the conditions are necessary. It can be shewn, as in Art. 55, that they are sufficient. 76. To obtain the forces of constraint at the fixed points in any of the cases considered in the last three propositions, we have only to apply the remaining conditions of equilibrium found in Chapter II. 77. As we shall often have to consider the case of bodies, such as rods, which are connected by means of hinges or joints, it will be well to consider what a hinge is. We shall consider smooth hinges only. The connection may be supposed to be made in several ways. A point of one body may be connected with one of the other body by a very short string. Or one body may end in a very small ball or pivot, which works in a corresponding small socket or ring in the other body, so that there is contact at only one point. Or we may suppose each body to end in a small ball, which works in a corresponding socket of a small separate body. In each of these cases there is no restriction on either body, except that the two ends must be in contact; the action on each at the common point must pass through this point, but will adapt itself in magnitude and direction so as to maintain equilibrium, if possible. If three or more bodies are connected by one joint, we may suppose the connection to be made by each having a very short string attached to it, and the strings to be knotted together. Or we may suppose each to end in a small smooth ball, which works in a corresponding socket in a small separate body. 78. In the construction of materials it is often desirable to ascertain the internal forces between one portion of a body and the adjacent portion. When all these are known, we are able to adapt the strength of each part to the force it has to sustain. For instance, if we know that the tension at one point of a chain is |