108 Observations on Inclined Planes. 



tance of attrition and rolling unaffected by Inertia. To do this, he 

 introduces into his equations the weight of the wheels separately 

 from the weight of the load, the wheels being increased in the ratio 

 of SG to SO, SG being the radius of the wheel, or the distance of 

 the center of gravity of the wheel from the circumference, and SO 

 being the distance of the center of oscillation from the same. This 

 latter point was found by causing the wheel to vibrate from a point 

 in the circumference, and calculating from the times of vibration. 

 Although this was necessary in Mr. Wood's experiments, in order to 

 obtain separately the amount of rubbing and rolling friction, yet it 

 will readily be seen that many cases will occur in practice, where it 

 will be inconvenient and sometimes impossible to ascertain the value 



SO 



of cp. In the equations here given, the inertia is included in the 



general estimate of friction, for the sake of convenient application. 



Examples will be given hereafter. 



Let V=velocity, and F= accelerating force, we always have, %y 



V2 2s 

 the laws of descent, ^"=^7, hence the equation above, t^=— 



v'^ 2s v^ 



becomes ,—, r-=— — from which x=sr'—-^ and substitutine; 



for &' Its value, ?z= , , ^ — C. 



^ v^ -\-2sx 



This equation may be used when the height of a plane of a given 

 length is required, such that a car, descending by its own gravity, 

 may acquire a given velocity at the foot of the plane ; or when the 

 height is given, the length s may be found. The value of x is found 

 from equation A by experiment. According to the experiments of 

 Mr. Wood, a;=.15, where the friction is equal to about the 220th 

 part of the weight. When the friction is taken as the 300th part of 

 the weight, as is sometimes done, a;-=.107. In general, if the fric- 



tion is equal to the mth part of the weight, we always have x=- . 



The friction f of the wheels being found, it is to be used as a 

 given quantity in the case where a train of cars, descending an in- 

 clined plane, draws up another train at the same time, the trains be- 

 ing connected by a rope passing over a wheel at the top of the plane, 

 and supported by small wheels or sheaves along the rail track. In 

 this case it is necessary to find the friction of the rope, sheaves, and 

 wheel at the head of the plane. 



