162 WORK 



To obtain the relation between d0 and dx we differentiate equation (a), 

 obtaining 



- 2(a - x)dx + I* sin - cos -cZ0 = 0. 

 2 2 



Thus G = w 



de 



. e e 



wl 2 sin - cos - 

 _ 22 



2(a -x) 



wl 2 sin 



4 A /a 2 - Z 2 sin 2 - 

 giving the couple required. 



EXAMPLES 



1. A square ABCD is formed by joining four equal rods by freely moving 

 hinges. The points A, C are joined by an elastic string of natural length equal 

 to a diagonal of the square, and of modulus X. What forces must be applied 

 to the points 5, D to stretch the string to 1 times its length ? 



2. Three spheres each of radius a and weight w are tied to a point P by 

 strings of natural length I and modulus X, and hang freely, touching each other. 

 Find the depth of their centers below P. 



3. The mechanism by which a Japanese umbrella is opened is such that each 

 rib turns through an angle of 5 for every inch that the sliding piece is moved 

 up the stick. If there are 18 ribs, each of weight i ounce, and having their 

 centers of gravity 10 inches from their pivots, find with what force the sliding 

 piece must be pushed up the stick to open the umbrella, when the stick is held 

 vertically, and the ribs are inclined at an angle of 30 to it. 



4. The hands of a clock are balanced with counterpoises, so as to be in 

 equilibrium in any position. When the time indicated by the clock is 5.10, a 

 bird of weight w suspends itself from a point on the minute hand which is six 

 feet from the pivot. How large a vertical thrust must be applied to the hour 

 hand, also at a point six feet from the pivot, to restore equilibrium ? 



5. A clock is wound by raising a weight of 20 pounds through a distance of 

 3 feet, this enabling the clock to run for 30 hours. The pendulum and escape- 

 ment are removed, so that the hands will "race" unless held fast. How large 

 a couple must be applied to the minute hand to prevent this occurring ? 



6. The coupling between two English railway carriages consists of a rod with 

 a right-handed and a left-handed screw cut at its opposite ends and turning 

 in nuts attached to the carriages. If the pitch of each screw is one inch, and 

 the rod is turned by a force of 56 pounds acting at best advantage at the 

 end of a lever 15 inches long, find the force by which the carriages are 

 drawn together. 



