196 G. F. Becker — Impact Friction and Faulting. 



except for moderate differences of velocity. A great advance 

 towards the elucidation of the true relation between friction and 

 velocity might evidently be made by a study of the relation 

 between the coefficient of restitution and the velocity of imper- 

 fectly elastic impinging spheres or cylinders. 



Effect of extreme velocities. — If there is relative translation be- 

 tween rubbing masses there must always be more or less 

 " play," or a slight possible movement normal to the frictional 

 contact, so that the centers of inertia of such masses cannot be 

 confined to motion in absolutely straight lines. This is of 

 importance in the extreme cases of zero velocity and ver}' 

 great velocit}-. The coefficient of friction for rest appears from 

 experiments to be merely the maximum value of the coefficient 

 for motion, and it s excess is almost beyond a doubt due in part 

 to the fact that, when at rest under a given pressure, the pro- 

 jections of the opposing surfaces interlock more deeply than 

 they can under the same pressure when in motion. In the 

 case of high velocities the mean distance of the centers of in- 

 ertia of rubbing masses must be greater than for low velocities, 

 owing to the increase of the elastic rebound, resolved perpen- 

 dicularly to the contact plane, when the more or less irregular 

 inequalities impinge.* These will therefore interlock to a 

 smaller extent, and the friction will diminish from this cause 

 independently of an}' variation of the coefficient of restitution. 

 This coefficient also stands in some not precisely known inverse 

 relation to the velocity so that friction should be expected to 

 be notably less for very high velocities than for moderate ones 

 as experiments have shown is the case. 



Results for a system of sheets. — If then a system of material 

 sheets is arranged say in a vertical pile, there being perfect contact 

 throughout, and impulsive energy is imparted to the uppermost 

 of them, this energy according to the deductions drawn will be 

 distributed throughout the system exactly as in a series of par- 

 tially elastic spheres. Assuming that the coefficient of restitu- 

 tion is constant and not unity, every sheet which receives 

 energy must move and must impart kinetic energy to the next 

 following. The distribution of energy must be logarithmic 

 (simply or compositely) quite irrespective of any differences of 

 frictional resistance which may exist at different contacts. On 

 the other hand if the friction at all contacts is the same, the 

 movements of the sheets will be logarithmic and if the friction 

 varies according to any law, this combined with the law of the 

 expenditure of energy will give the locus. Morin's law that 



* Imagine for example a mass of india-rubber dragged over a street pavement ; 

 the elastic mass would constantly rebound from the irregular paving stones, and 

 the more rapid the translation the higher would be the mean position of the 

 center of inertia of the rubber mass. Morin shows that the resistance offered by 

 pavement to a vehicle provided with springs is less than to one without springs. 



