ON THE RESISTANCE, OF ROAD VEHICLES TO TRACTION. 319 



effort, or the pull exerted by the horse. This value corresponds to values 

 determined from : — 



The inclination of the ground . « . • (0 



The inclination of the pull . . . • («) 



The weight of the shaft and its additional load (P) 



The weight of the frame and pole . i . (p) 



The total rise of the ground during the run . {h) 



The distance run . t • . i . (L) 



The coefficient .«•.«. (y') 



Now sin (i) = — , and the total pressure transmitted to the ground 



Li 



^(P+p). Applying the theorem of work to the movement of the 

 vehicle : — The force F acts through a distance L, and its direction makes 

 with the surface of the ground an angle (a). 



Therefore the work done in moving the distance li is equal to 

 FL cos o. 



Work due to resistance is composed of : — • 



(i.) That due to the weight of the vehicle, iSicjj and \^hich is equal to 

 the weight multiplied by the total rise. 



Work due to Weight = (P+jo)/i. 



(ii.) That due to the reaction of the ground upon the wheel : — 

 Calling E, that reaction or resistance to rolling, which acts tangentially to 

 a wheel whose radius is r, and if the tyre develops itself exactly upon 

 the road, the point of application of the force travels a distance L, then 



Work due to the reaction of ground on wheel (R) = B,L. 



(iii.) That due to the friction of the axle in its bearings : — The spindle 

 is acted upon by two forces, i.e. the pull F, which makes an angle of 

 (a + t) with the horizon, and the weight or vertical force (;*). The re- 

 sultant of these two forces is normal to the circumference of the axle, and 

 is represented by the third side of a triangle, of which F and P are the 



other two, making between them an angle of ( '^ + a + i ) 



.'. Resultant =v'FM^P2^2"F^sinl;M^i) 



and if we multiply this expression by the coefficient of friction (/) we 

 obtain the tangential rolling force applied to the spindle. The distance 

 travelled by its point of application is equal to the distance (L) run by 

 the wheel multiplied by the ratio of the radius of the spindle (p) to radius- 

 of the wheel (/•). 



Then the work due to the friction of the axle in its bearings 



= L ^ s/Y^+pi^ 2'Fp~sm(c[+i) 



The velocity of the vehicle was kept uniform throughout the experi- 

 ments, and the variation of the inomentum of the system was zero. 

 The equation of Work becomes — 



FL cos a =t: ±(P +p)h + RL + ^ VF^ + P2-2rpsin(o + r) 



