74 STATICS OF SYSTEMS OF PARTICLES 



13. If the weight of the wheels and axles of a car is to, and if this may not 

 be neglected in comparison with W, the total weight of the car, show that 

 equation (c) of example 5, p. 72, must be replaced by 



_ ( W w) b sin e 



14. A locomotive of weight 134 tons rests on a bogie, of which the wheels and 

 axles weigh 4 tons, and two pairs of driving wheels, of which the wheels and 

 axles weigh 10 tons. The weight taken on the axles of the bogie is 40 tons, 

 that taken on the axles of the driving wheels being 80 tons. The diameters of 

 the wheels are 2 feet 10 inches and 7 feet 1 inch respectively. Each axle, where 

 it passes through the axle box, is of radius 2 inches, and the coefficient of fric- 

 tion is A. Find the horizontal force necessary to move the engine. 



15. In question 14 the rails are greased so that the coefficient of friction 

 between them and the wheels is less than yi^; show that the engine cannot 

 be started without the wheels skidding on the greased rails, and explain the 

 dynamical processes by which the engine is set in motion in this case. 



STRINGS 



53. Strings, ropes, and chains frequently form part of the systems 

 of bodies with which statical problems are concerned, so that it is 

 important to discuss the equilibrium of a string (or rope or chain). 

 The first problem we shall consider is that of a string stretched 

 over a surface, as for example a pulley wheel, it being supposed 

 that the weight of the string may be neglected, and that the con- 

 tact between the string and the surface is equally rough at all 

 points. It will also be supposed that the string is all in one plane. 



Let P, Q be two adjacent points of the string so near together 

 that the portion PQ of the string may be treated as a particle. 



The forces acting on this particle will be 



(a) T pt the tension at P, acting along the tangent to the string 

 at P; 



(b) T Q , the tension at Q, acting along the tangent to the string 

 atC; 



(c) the reaction with the surface. 



By Lami's theorem, each force must be proportional to the sine 

 of the angle between the remaining two forces. 



