forces acting on a rigid body may be replaced by a single force acting through the centre of inertia of the body, and which is alone effective in producing motion of translation, and a couple which is alone effective in producing motion of rotation. For, taking any two of the forces which are not equal and opposite parallel forces, we may replace them by their resultant. This resultant can then be combined with one of the remaining forces, and so on till finally we have left either (1) a single force passing through the centre of inertia, when translation only takes place ; or (2) two equal and opposite parallel forces, which produce rotation only; or (3) a single force which does not pass through the centre of inertia. In this last case, if we add two equal and opposite forces acting through the centre of inertia and parallel to the resultant, they will not influence the motion. One of these forces will then form a couple with the resultant, and the other will be a force equal and parallel to the resultant, acting through the centre of inertia and tending to produce translation of the body. - P.BE+P/12 CHAPTER IX EQUILIBRIUM OF FORCES 71. Equilibrium.-When the forces which act on a body are so balanced that they produce no acceleration in the body, that is, do not alter its state of motion, they are said to be in equilibrium. A study of the conditions that have to be fulfilled in order that the forces considered may be in equilibrium is sometimes considered as a separate branch of mechanics, called Statics. The name statics is at first sight rather misleading, since it does not follow because the forces acting on a body are in equilibrium that the body is at rest, for if the body is originally moving the velocity will continue uniform, and not be altered by the forces. The appropriateness of the name, however, is apparent, if we consider that unless the forces acting on a body are in equilibrium it is impossible for the body to remain at rest. Hence we may if we like define forces in equilibrium as such that they render it possible for the body on which they act to remain at rest. 72. Conditions for Equilibrium of a Particle.-It is obvious that a particle acted upon by a single force cannot be in equilibrium. For two forces acting on a particle to be in equilibrium, they must fulfil the following conditions: They must be (1) equal in magnitude, (2) act along the same straight line, (3) be of opposite sense. When referring to these conditions in future, we shall for shortness simply say that the forces must be equal and opposite, but it must be remembered that this is only an abbreviation for the above three conditions. The condition that three forces acting on a particle may be in equilibrium is that any one of the forces must be equal and opposite to the resultant of the remaining two, for we may, if we please, replace any two of the forces by their resultant, when we should have reduced the problem to the equilibrium of two forces. The resultant of any two of the forces, say P and Q, must lie in the plane containing P and Q. Hence if there is to be equilibrium the third force, since it must be equal and opposite to this resultant, must also lie in the plane containing the other two forces. Hence the first condition for equilibrium is that the three forces must all lie in one plane. As to the relations between the magnitude of the forces, the resultant of any two (P and Q) is represented by the diagonal OR (Fig. 54) of the parallelogram constructed on the lines OP and OQ as adjacent sides. Hence the other force must be represented in magnitude and direction by RO, or by os, where os is equal to OR and in the same straight line with it. Since QR is equal to OP, we may take QR to represent the force P in magnitude and direction (§ 65). Then the three forces will be represented by OQ, QR, and RO, the sides of a triangle. Hence if it is possible to draw a triangle of which the sides are parallel or perpendicular to the three forces and proportional to them in magnitude, the forces will be in equilibrium. It must be specially noticed that in drawing the triangle the sides must all be drawn in the same sense as the forces, so that when we place arrows on the sides to show in which sense the forces act, all the arrows may point the same way round the triangle, as shown at A B C in Fig. 54. The conditions of equilibrium for any number of forces acting on a particle are that the forces can be represented in magnitude and direction by the sides of a closed polygon taken in order, i.e. drawn in the same sense as the forces. This at once follows from the polygon of forces, for the resultant of all the forces but one is represented in magnitude and direction by the line joining the starting-point to the end of the last line drawn in the polygon, i.e. by the remaining side of the polygon, which by supposition represents in magnitude the only force not yet included, but is in an opposite sense. 73. Conditions for Equilibrium of a Rigid Body. In the case of a rigid body the lines of action of the forces need not all pass through a single point, and in order that the body may be in equilibrium the forces must not tend to produce either translation or rotation. If the directions of all the forces pass through a single point they cannot produce rotation, and hence if they fulfil the conditions given in the preceding section for a particle they will be in equilibrium. If, however, the lines of action of the forces do not all pass through a point, then, in order that there may be no rotation, they must have no resultant moment tending to turn the body about any axis. The general condition for equilibrium is therefore that the sum of the moments of all the forces F taken about every point must be zero, and that the forces can be represented in magnitude and direction by the sides of a closed polygon taken in order. Since in most cases we shall only have to deal with forces acting in a plane, it is of interest to examine the condition for equilibrium in this case a little more fully. As by supposition the forces all act in a plane, it is evident that they can only tend to produce motion in this plane (by Newton's second law). Hence if we take two fixed lines not parallel (and preferably at right angles) in this plane, every possible translation must either be parallel to one or other of these lines, or else compounded of translations parallel to the two. Hence if the sum of the components of the forces when resolved parallel to these lines is zero, there will be no tendency to motion along either of these directions, so that there will be no translation. The condition for no rotation is that the sum of the moments about every point in the plane shall be zero. If both conditions are fulfilled there is equilibrium. If only the first condition holds, then there is rotation without translation, i.e. all the points of the body move in circles about a fixed point as centre; if the second condition alone is fulfilled, then there is translation without rotation, ¿.e. all the points of the body move with the same velocity in parallel paths. CHAPTER X WORK AND ENERGY 74. Definition of Work.-When a force acts upon a body, and the point of application of the force moves in the direction of the line of action of the force, the force is said to do work on the body. The amount of work done by the force is measured by the product of the force into the distance, measured along its line of action, moved through by its point of application. Hence if a force F acts on a body while its point of application moves through a distance s in the line of action of the force, the work (W) done by the force is given by W= Fs. If the body moves through a distance s in the direction opposed to the force, work is said to be done against the force, the work done being as before measured by the product Fs. If the displacement of the point of application of the force is not along the line of direction of the force, but inclined to it, then we must calculate the component of the displacement along the direction of the force, and this component multiplied by the force gives the work done either by or against the force, as the case may be, during the displacement. Thus suppose AC (Fig. 55) represents the direction of the line of action of the force (/) and AB the displacement of the point of application. Then the component of the displacement along the line of action of the force is C B AD, obtained by drawing BD perpendicular to AC. Hence the work done is F.AD. The correctness of the above construction is evident, for the displacement AB of the point of application may be replaced by the dis FIG. 55. placements AD and DB. During these displacements, no work will be done by the force while the point of application is moving from D to B, since the movement is at right angles to the line of action of the force. If we call the angle between the line of action of the force and the line of |