STATICS. COMPOSITION AND RESOLUTION OF FORCES. 25. The Composition of Forces is the operation by which the resultant of any number of forces, applied to the same point or body, is determined. 26. The Resolution of Forces is just the inverse of the composition; and it is sometimes called the Decomposition of Forces. 27. If two forces act upon a body in the same direction, the combined effect is equivalent to the sum of the forces. Thus, if a force of 4 pounds, and another of 5 pounds, act upon the same body in the same direction, the body will be acted on, or drawn, by a force of 9 pounds. But if two forces act in opposite directions, the resultant will be the difference of the two, and in the direction of the greater. Thus, if a body be acted on by a force of 8 pounds in one direction, and 6 pounds in the opposite, it will be drawn by a force of 2 pounds in the direction of the greatest force. 28. If a body at A (Plate I. Fig. 1) be acted upon se parately by two forces in the directions A B and A C, which would cause the body to be carried through the spaces A B and A C in the same time, then both forces acting together will cause the body to describe the diagonal A D of the parallelogram A B C D, in the same time in which it would describe either of the sides by either of the forces acting separately. If the forces at A cause the body to move uniformly along the lines A B, A C, then, since the force acting in direction A C parallel to B D, by the second law of motion, will not alter the velocity of the body towards the line B D, the body will therefore arrive at B D in the same time, whether the force in direction A C be impressed or not; therefore, at the end of the time, it will be found somewhere in the line B D. By the same argument, it will be found somewhere in the line CD: therefore it will be found in D, their point of intersection; and, by the first law of motion, it will move in a right line from A to D. Otherwise, Suppose the line A C to move parallel to itself into the place B D, whilst A moves from A to C, then, since this line and the body are both equally moved towards B D, it is evident that the body will always be in the moveable line A C: therefore, since the motions are uniform, and the lines A B, A C, are described in the same time, the body will describe the straight line A D, the diagonal of the parallelogram. But if the body A (Fig. 2) be carried through A B by a uniform force in the same time that it would be carried through AC by an accelerative force, then, by both forces acting together, it will, at the end of that time, be found in the point D, as before; but, in this case, it will describe a curve line A FD. Cor. 1. The forces in the directions A B, A C, A D, are respectively proportional to the lines A B, A C, A D. (Fig. 1.) Cor. 2. The two oblique forces A B and A C, are equivalent to the single and direct force A D, which is compounded of these two, by drawing the diagonal of the parallelogram. (Fig. 3.) Cor. 3. If two forces, as A B and A C, act in directions A B and A C respectively, then, because the diagonals of a parallelogram bisect each other, the forces represented by A B and A C are equivalent to twice the force represented by A E. Cor. 4. A given force may be resolved into two others, one of which is given and has a given direction. 29. If a body be kept in equilibrio by the joint actions of three forces in the same plane, these forces will be respectively proportional to the three sides, A B, B C, A C, of a triangle, which are drawn parallel to the directions of the forces DA, E A, A C. (Fig. 4.) Let A C represent the force C, and produce DA, E A, and complete the parallelogram ABC F. Now, by the last Prop. the force A C is equivalent to the two forces A B, A F; put, therefore, the forces A B, A F, instead of A C, and all the forces will still be in equilibrio: therefore, since AC represents the force C, then A B will represent its opposite force D, and B C or A F its opposite force E. Consequently, the three forces D, E, C, are proportional to A B, B C, A C, the three sides of the triangle A B C, formed by drawing lines parallel to the directions of the three forces. Cor. 1. The three forces D, E, C, will be respectively as the sines of the angles A C B, C A B, A B C; for these forces are as A B, B C, A C, and these sides are as the sines of their opposite angles C, A, B. Cor. 2. Three forces acting upon a body, and keeping it in equilibrium, are proportional to the sides of a triangle formed by drawing lines either perpendicular to the directions in which the forces act, or making any given angles with those directions. For such a triangle is always similar to that which is made by drawing lines parallel to the directions. 30. If three forces, the directions of which concur in one point, are represented by the three contiguous edges of a parallelopiped, their resultant will be represented, both in magnitude and direction, by the diagonal drawn from the point of concourse to the opposite angle of the parallelopiped. Let the directions in which the forces act be A B, A C, A D, and A G, (Fig. 5) and complete the parallelopiped BF. Then, since A B H C is a parallelogram, the force A H is equivalent to the two forces A B, A C; but D G is both equal and parallel to A H, and A D is both equal and parallel to G H; therefore A D G H is a parallelogram, and a force which is represented by its diagonal A G, equivalent to the two forces represented by A D, D G, that is to the three forces A B, A C, A D. is Cor. 1. If four forces, in different planes, act upon à body and keep it in equilibrio, these forces are to each other as the three edges and diagonal of a parallelopiped, constructed upon lines respectively parallel to the directions of the forces. Cor. 2. It also follows, that a single force may be resolved into three others in different planes: also, each of these forces may again be resolved into others, either in the same or different planes; and so on, as far as we please. 31. The properties in the preceding propositions hold good for all similar forces whatever, whether they be instantaneous or continual, or whether they act by percussion, drawing, pushing, pressing, or weighing, and are of the utmost importance in mechanics and the application of the doctrine of forces to natural philosophy. Example 1. Suppose a boat to be fastened to a fixed point by a rope, and acted on at the same time by the wind and the current; then the direction of the rope will represent the resultant of these forces. Thus, let A B and A C (Fig. 6) be at right angles, and the forces in these directions 30 pounds and 40 pounds respectively; required the magnitude and direction of the resultant. Since the angle C A B is a right angle, and A B = 30, AC 40, A D will be the resultant; and A D2 = 30% 2500 .. A D 50 pounds, the magnitude of the resultant. And to find the angle D A C, we have, DC А В 30 3 +40 sin. D A C = = = = 6. By the table AD AD 50 5 of sines, 6 is the sine of 36° 52'. C Example 2. If two equal forces act at an angle of 120°; prove that the resultant or force compounded of them is equal to either of the equal forces. Let A B and A D (Fig. 7) represent the two equal forces, and the angle B A D = 120°; then draw B C parallel to A D, and D C parallel to A B. Join A C; and since the angle B A D = 120°, the angle A B C, which is equal to the supplement of the angle B A D, must be equal to 60°; but A B = B C, therefore the angle B A C will be equal to the angle B C A. And since the three angles of the triangle ABC, taken together, are equal to two right angles, or 180°, we have 60° +LBAC+LBCA 180° ..LBAC=120° ÷ 2 = 60° Therefore since each angle of the triangle ABC is equal to 60°, the triangle is equilateral, or A B and B C are each equal A C. 32. We may very frequently see examples of the resolution of forces, where the force exerted being resolved into two, one of them is totally lost or counteracted, and the remaining part only is effective. Thus, when we draw any body along the ground by a rope fastened to it, (Fig. 8), and supposing the rope to be inclined to the horizon at an angle of 45 degrees, the force which we exert is effective only in part. Thus, if we exert a force of 20 pounds, this is equivalent to two; one in the direction of A B, perpendicular to the horizon, and the other in the direction B C, parallel to the horizon; and A B+B C2 = 20; but since the angle A C B = 45°, A B = B C .. 2 B C2 = 20o or B C2 = 4002, therefore B C = 200 1414 pounds. Hence the force with which we draw the body horizontally is 14.14 pounds. |