68 STATICS. [117. 



(7) Explain how the centroid of a plane area can be found graphi- 

 cally by dividing the area into narrow parallel strips. 



(8) A homogeneous rectangular plate is pivoted on a horizontal axis 

 through its centre so as to turn freely in a vertical plane. If weights 

 W^ Wo_, Wft W be suspended from its vertices, what is its position of 

 equilibrium ? 



(9) The ends of a straight lever of length / are acted upon by two 

 forces FI, F 2 in the same plane with it, but inclined to the lever at angles 

 !, 2 . Determine the position of the fulcrum. 



117. Funicular Polygons and Catenaries. The funicular polygon 

 in its original meaning represents the form of equilibrium 

 assumed by a string or cord suspended from two fixed points 

 and acted upon by any forces in the same plane. The "cord" 

 is supposed to be perfectly flexible, inextensible, inelastic, and 

 without weight. When the number of forces is made infinite, 

 the polygon becomes a continuous curve called a catenary. 



The present discussion is confined to the case when the 

 forces are all vertical so that they can be regarded as weights. 



118. Let A, B (Fig. 26) be the fixed points, and let there be 

 five weights, l v W^, W%, W^ W%> suspended from the points 

 I, II, III, IV, V, of the cord. 



If the cord be cut on both sides of the point I and the corre- 

 sponding tensions T v T 2 be introduced, the point I must be 

 in equilibrium under the action of the three forces W^, T v 7" 2 . 

 Hence drawing a line I 2 to represent the weight W and draw- 

 ing through its ends I, 2 parallels to AI and I II, respectively, 

 we have the force polygon of the point I. Its sides O I and 2 O 

 represent in magnitude, direction, and sense the tensions 7\, 

 T 2 ; in other words, the weight W-^ has thus been resolved into- 

 its components along the adjacent sides. 



The same can be done at every vertex of the polygon 

 I II III IV V, and all tensions can thus be found. But as the 

 the tension 7^ in I II occurs again (with sense reversed) in the 

 force polygon for the point II, and so on, the successive 

 force polygons can be fitted together, every triangle having one 



