EXAMPLES 111 



This equation enables us to trace the changes in the value of the angle 6 as 

 E B is gradually increased. We find that when R B = the value of is 6 = 0, 

 and that as R B increases increases continually, but never exceeds the value 

 = a, which is reached when Rj$ = oo. . 



If e < a, the value of 6 will pass through the value e when Eg reaches a 

 certain value, namely 



Wsine 



RB = - ; - - 

 sin (a e) 



and slipping at A will take place at this point. 



If c > a, the value of 8 will never reach the value e, so that slipping at A can 

 never occur. Thus equilibrium is never broken, and the harder we press at B 

 the more firmly the ring is held between the finger and the table. 



We can now summarize the results which have been obtained, as follows : 



If a > e', the ring rolls along the table as soon as we begin to press at B. 

 If a < e', there are two cases : 



(a) a > e, the ring will slip at A as soon as sufficient pressure is applied ; 

 (6) a. < e, the ring cannot be made to move under any amount of pressure. 



To make the ring shoot out from under the finger by slipping at A (in which 

 case it returns to the hand, as in the well-known trick), it is necessary to press 

 at a point on the ring at which a is greater than e, while being less than e'. 

 We notice that if c is greater than e', it is impossible to project the ring in this 

 way ; this can only be done if the contact with the finger is rougher than the 

 contact with the table. 



GENERAL EXAMPLES 



1. A pair of steps is formed by two uniform ladders each of length 12 

 feet and weight 20 pounds, jointed at the top, and having their points at 

 distances 5 feet from the ground connected by a rope. The steps stand on 

 a smooth horizontal plane, and a man of weight 160 pounds ascends to a 

 height of 9 feet on one side. Find the tension in the rope. 



2. A heavy uniform rod is supported by two strings of lengths a, b. 

 The upper ends of the strings are tied to the same point, the lower ends 

 being tied to the two ends of the rod. Show that the tensions of the strings 

 are proportional to a and b respectively. 



3. Two small fixed pegs are in a line inclined at an angle 8 to the 

 horizon. A rough thin rod passes under the lower and rests on the higher, 

 this latter being lower than the center of gravity of the rod. The distances 

 of the center of gravity from the two pegs are a and b respectively, and the 

 coefficient of friction is /*. Show that if the rod is on the point of motion, 



b + a 



