EXAMPLES 87 



4. A weightless string 24 inches in length is fastened to two points 

 which are in the same horizontal line, and at a distance of 16 inches apart. 

 Weights are fixed to two points at distances of 9 and 7 inches from the 

 ends of the string and hang in such a way that the portion of the string 

 between them is horizontal. Determine the ratio of the weights. 



5. A light wire has a weight suspended from its middle point, and is 

 itself supported by a string fastened to its two ends and passing over a 

 smooth peg. Show that the wire can rest only in a horizontal or vertical 

 position. 



6. Three smooth pegs A, B, C stuck in a wall are the vertices of an 

 equilateral triangle, A being the highest and the side BC horizontal ; a 

 light string passes once around the pegs and its ends are fastened to a 

 weight W which hangs in equilibrium below BC. Find the pressure on 

 each peg. 



7. Two rings of weights P and Q respectively slide on a weightless 

 string whose ends are fastened to the extremities of a straight rod inclined 

 at an angle to the horizontal. On this rod slides a light ring through 

 which the string passes, so that the heavy rings are on different sides of 

 the light ring. All contacts are smooth and, in equilibrium, <f> is the angle 

 between the rod and those parts of the string which are close to the light 

 ring. Prove that tan ^ p -Q 



tan ~ P + Q ' 



8. Two small heavy rings slide on a smooth wire, in the shape of a 

 parabola with axis horizontal ; they are connected by a light string which 

 passes over a smooth peg at the focus. Show that their depths below the 

 axis are proportional to their weights when they are in equilibrium. 



9. Two equally heavy rings slide on a wire in the shape of an ellipse 

 whose major axis is vertical, and are connected by a string which passes 

 over a smooth peg at the upper focus. Show that there are an infinite 

 number of positions of equilibrium. 



10. ABCD is a quadrilateral; forces act along the sides AB, BC, CD, 

 DA measured by a, /3, 7, 8 times those sides respectively. Show that if 

 these forces keep any system of particles in equilibrium, then 



ay = /35. 



11. A light rod rests wholly within a smooth hemispherical bowl of 

 radius r, and a weight W is clamped on to the rod at a point whose dis- 

 tances from the ends are a and &. Show that 0, the inclination of the rod 

 to the horizon in the position of equilibrium, is given by the equation 



2 Vr 2 - ab sin = a - 6. 



