99, 100] PURE ROLLING. 307 



The sliding the ceases. Obviously it cannot change sign, so that the 

 above solution ceases to hold. The ball now rolls without sliding, 

 and we have always, at subsequent instants, the equations of constraint 



dx dy 



Y = &y _ n *p_ _ <* V 



M dt* dt " A 



From this we obtain 



so that X = Y = 0, and the ball moves uniformly in a straight line. 

 In reality there is always a certain friction of pivoting, causing 

 a moment about the normal, but this would only affect the rotation 

 component r, which would not affect the motion of the center of 

 the ball. 



1OO. Pure Rolling. The preceding problem has illustrated 

 both sliding friction and pure rolling. The treatment of the latter 

 is interesting on account of a peculiarity in the nature of rolling 

 constraint which makes the ordinary treatment of Lagrange's equations 

 require modification. We shall accordingly first present the application 

 of Euler's equations to this subject, but before doing so, we will 

 treat by means of results already obtained one of the most important 

 practical problems, which illustrates the steering of the bicycle, namely 

 the rolling of a hoop or of a coin upon a rough horizontal plane. 



As the hoop rolls, if its plane is not vertical, it tends to fall, 

 and thus to change the direction of its axis of symmetry. The falling 

 motion developes a gyroscopic action, which causes the hoop to pivot 

 about the point of contact, so that the path described on the table 

 is not straight but curved. The pivoting motion, like the precession 

 of the top, tends to prevent the falling, and to this is added the 

 effect of the centrifugal force due to the curvilinear motion of the 

 center of mass. Thus the hoop automatically steers itself so as to 

 prevent falling, and a bicycle left to itself does the same thing. 



Let the position of the hoop be defined by the coordinates of 

 its center of mass, and by the angles #, ^,9 of 90, # being the 

 inclination to the vertical of the axis of symmetry, or normal to the 

 plane of the hoop at its center. We will examine the conditions for 

 a regular precession, in which <$, qp', ^' being constant, the center of 

 mass and the point of contact of the hoop with the table evidently 



20* 



