The arms are assumed to be of rectangular section; if they are not, the error involved will be exceedingly small. The sphere may be regarded as being made up of a great number of thin circular layers of radius r1, and radius of gyration (see p. 96). = R R5 CHAPTER IV. RESOLUTION OF FORCES. WE have already explained how two forces acting on a point may be replaced by one which will have precisely the same effect on the point as the two. We must now see how to apply the principle involved to more complex systems of forces. Polygon of Forces. If we require to find the resultant of more than two forces which act on a point, we can do so by finding the resultant of any two by means of the parallelogram of forces, and then take the resultant of this resultant and the R/234 next force, and so on, as shown in the diagram. The resultant of 1 and 2 is marked R1.2., R2 and so on. Then we finally get the resultant R1.2.3.4. for the whole system. Such a method is, however, clumsy. The following will be found much more direct and convenient: Start from any point O, and draw the line i parallel and equal on a given scale to the force 1; from the extremity of 1 draw the line 2 equal and parallel to the force 2; then, by the triangle of forces, it will be seen that the line R1.2. is the resultant of the forces 1 and 2. From the extremity of 2 draw 3 in a similar manner, and so on with all the forces; then it will be seen that the line R1.2.3.4. represents the resultant of the forces. In using this construction, there is no need to put in the lines R1.2., etc.; in the figure they have been inserted in order to make it 0 R1234 FIG. 128. clear. Hence, if any number of forces act upon a point in such a manner that lines drawn parallel and equal on some given scale to them form a closed polygon, the point is in equilibrium under the action of those forces. This is known as the theorem of the polygon of forces. a d Method of lettering Force Diagrams.-In order to keep force diagrams clear, it is essential that the forces be lettered in each diagram to prevent confusion. Instead of lettering the force itself, it is very much better to letter the spaces, and to designate the force by the letters corresponding to the spaces on each side, thus: The force separating a from b is termed the force ab; likewise the force separating d from b, db. с FIG. 129. This method of notation is usually attributed to Bow; several writers, however, claim to have been the first to use it. Funicular or Link Polygons. When forces in equilibrium act at the corners of a series of links jointed together at their extremities, along each link can be readily found by a special application of the triangle of forces. Consider the links ag and bg. There are three forces in equili a b FIG. 130. brium, viz. ab, ag, bg, acting at the joint. The magnitude of ab is known, therefore the magnitude of the other two acting on the links may be obtained from the triangle of forces shown on the right-hand side, viz. abg. Similarly consider all the other joints. It will be found that each triangle of forces contains a line equal in every respect to a line in the preceding triangle, hence all the triangles may be brought together to form one diagram, as shown to the extreme right hand. It should be noticed that the external forces form a closed polygon, and the forces in the bars are represented by radial lines meeting in the point or pole g. It will be evident that the form taken up by the polygon depends on the magnitude of the forces acting at each joint. |