304 VII. DYNAMICS OF ROTATING BODIES. 



the point of contact. If it is in the tangent plane, the motion is 

 said to be pure rolling, and the bodies act as if "perfectly rough". 

 If the instantaneous angular velocity has a normal component, this 

 is known as pivoting, and is also resisted by a frictional moment. 

 The pivoting friction is however usually neglected where the surfaces 

 are supposed to touch at a single point. The conception of perfect 

 roughness, involving the absolute prevention of slipping under all 

 circumstances is as far from the truth as that of perfect smoothness, 

 nevertheless slipping may often cease in actual motions, so that 

 motions of perfect rolling, whether or not accompanied by pivoting, 

 are important in practice. For instance, a bicycle wheel under 

 normal circumstances rolls and pivots, if it slips the consequences 

 may be serious. 



In the following sections we shall consider the methods of treating 

 various cases of friction. We may however, without calculation, 

 consider the effect of imperfect friction on the motion 

 of the top spinning on the table. Let P (Fig. 108) 

 represent the peg, no longer considered as a sharp 

 point. Let OH represent the angular momentum at 

 the center of mass 0. The friction is in the direc- 

 tion Fj opposite to the motion of the point of 

 contact of the peg with the table. The moment of 

 this force with respect to the center of mass is 

 perpendicular to the plane 01$ or K. Thus the end 

 of OH moves in the direction of K, that is rises. 

 Thus the effect of friction is to make the top rise 

 toward a vertical position. When it has reached 

 that, it "sleeps" and the friction has become merely 

 pivoting friction, tending to stop the motion. We 

 have before seen that under conservative forces, the top would never 

 become vertical except instantaneously by oscillation. 



The effect of friction on the Maxwell top may be most easily 

 seen from the fact that the friction tends to stop the spinning, 

 accordingly it causes a moment which is represented by a vector 

 opposite in direction to ro, Figs. 83 a, b. Compounding this vector 

 with H we see that the moment of momentum vector H tends to 

 move away from the axis of the two cones in Fig. 83 b while it tends 

 towards it in Fig. 83 a, thus the trace of the invarible axis (as it 

 would be but for friction), instead of being an ellipse, is a spiral 

 winding outwards in the former case, and inwards in the latter, as 

 is shown by the arrows in Fig. 83. 



99. Motion of a Billiard Ball. We wiU now treat the 

 problem of the motion of a sphere on a horizontal plane, taking 



