EXAMPLES 187 



17. A ladder of length h and weight w stands in a vertical position on 

 a rough floor, an elastic string being tied to its topmost point and to a 

 point in the ceiling at a height b above the floor, its tension being T. Show 

 that the equilibrium is stable or unstable according as 



b - 



18. If the tension in question 17 is equal to w(b h)/2b, determine 

 whether the equilibrium is stable or unstable. 



19. If in question 14 the rods are of the critical length which separates 

 stability from instability, show that the equilibrium is neutral, so that the 

 rods can, within certain limits, rest in any position in the plane perpen- 

 dicular to the axis of the cylinder. 



20. Show that the rods in the last question are in stable equilibrium 

 as regards displacements in which the plane of the rods rotates about a 

 vertical axis, and find the couple required to hold the rods in a position in 

 which the plane makes any given angle 6 with the axis of the cylinder. 



21. Find the smallest length of the rods in the last question, in order 

 that the equilibrium may be stable for all possible displacements. 



22. The radii of curvature at the blunt and pointed ends of a hard-boiled 

 egg are a and b respectively, and the egg can just be made to balance on 

 its blunt end when stood on a rough horizontal surface. Show that it can 

 be made to balance on its pointed end if stood inside a hemispherical basin 

 of radius less than 



where c is the longest axis of the egg. If the radius of the basin is just 

 equal to a(c b)/(a + b c), would the equilibrium be stable or unstable? 



23. A, B, C are three equidistant smooth pegs in the same horizontal 

 line, and a heavy uniform string has its ends tied to A, C, and is looped 

 over B. Show that there may or may not be a position of equilibrium in 

 which the two catenaries AB, EC are unequal, and that if there is such a 

 position it will be stable. 



Show also that the position of equilibrium in which the middle point of 

 the string is at B is unstable or stable according as an unsymmetrical 

 position of equilibrium does or does not exist. 



