54 



FORCES ACTING OK A SINGLE PARTICLE 



The region in which equilibrium is possible assumes two different forms 

 according as the angle of the inclined plane a is less or greater than the angle 

 of friction e. In the former case the region of equilibrium is of the kind repre- 

 sented in fig. 27. On passing the value a. = e the circle used in the construction 

 passes through the point O, and for values of a greater than e the region of 

 equilibrium becomes an area of the kind drawn in fig. 28. On passing through 

 the value a = e a sudden change takes place in the shape of the region of 

 stability. For values of a which are greater, by however little, than e, a circle 

 of radius Z, center 0, is entirely outside the region of stability ; while for values 

 of a which are smaller, by however little, than e, this circle is inclosed within 

 the region of equilibrium. Clearly this circle maps out the region within which 

 the weight can rest with the string unstretched, and this will be one of equi- 

 librium or not according as a < or > e. 



Thus this circle falls inside or outside the region of equilibrium in the way 

 predicted by analysis. At the same time we could not have been sure, without 

 a separate investigation, that the result given by analysis would be accurate as 

 regards the region within a distance I of O. For the analysis began by assuming 

 the string to be stretched, and so had no application except to the region at a 

 distance greater than I from 0. 



4. Two weights w, w f rest on a smooth sphere, being supported by a string 

 which passes through a smooth ring at 0, a point vertically above the center of the 

 sphere. Find the configuration of equilibrium. 



Let P, Q, in fig. 30, be the positions of the two weights 

 in a configuration of equilibrium. The weight w at P is 

 acted on by the following forces : 



(a) its weight w vertically downwards ; 



(6) the tension of the string along PO; 



(c) the reaction between the sphere and the weight. 

 Since the sphere is supposed smooth, the direction of 

 this reaction is at right angles to the plane of contact 

 between the particle and the sphere ; i.e. along CP. 



The three forces acting on the particle P are accord- 

 ingly parallel to the three sides of the triangle OPC. 

 Thus the triangle OPC may be regarded as a triangle of 

 forces for these forces, so that the magnitudes of the 

 forces must be proportional to the sides of this triangle. 

 Denoting the tension and reaction by T and J?, we obtain 



Fia. 30 



oc~ OP~ CP 



(a) 



In the same way the triangle OCQ may be regarded as a triangle of forces for 

 the particle Q. (This triangle does not represent force on the same scale as the 

 former triangle OCP ; for in the former case 0(7 represented a weight w, whereas 

 it now represents a weight w'.) 



