72 STATICS OF SYSTEMS OF PAKTICLES 



Now e, a, and 6 are supposed to be the same for each wheel, so that 

 6 sin e 



by equations (a) and (6). 



Thus P = , (c) 



giving the horizontal pull required. 



The value of 6, the radius of the axle, will generallyjDe small in comparison 

 with a, the radius of the wheel. Thus, without serious error, we may neglect 

 6 2 sin 2 e in comparison with a 2 , and replace the denominator in equation (c) by a. 

 The equation now becomes 



p _ Wb sin e 

 a 



By making b/a very small, we see that the car can be made to run very 

 smoothly. We notice also that even if there is so much friction between wheel 

 and axle that the coefficient of friction may be regarded as infinite, we have 

 sin e 1, and hence 



so that the force required to drag the car along will still be small compared 

 with that required to drag the same weight over a fairly smooth surface. 



This analysis has assumed that the wheels may be supposed to touch the 

 ground only at their lowest point. It applies pretty accurately to the case of 

 steel wheels rolling on steel rails, but does not apply to the problem of an 

 ordinary road carriage moving over a soft road, where the wheels are embedded 

 to a small extent in the road. In fact, if the analysis just given took account of 

 all the facts of the case, it is clear that the force required to haul a car would 

 be independent of the state of the road. 



EXAMPLES 



1. A weight of 250 pounds is suspended from a light rod which ;s placed 

 over the shoulders of two men and carried in a horizontal position. If the men 

 walk 10 feet apart and the weight is 4 feet from the nearer of them, find the 

 weight borne by each. 



2. A weight is suspended from a light rod which passes over two fixed sup- 

 ports 6 feet apart. On moving the weight 6 inches nearer to one support, the 

 pressure on that support is increased by 10 pounds. What is the amount of the 

 weight ? 



