232 PROFESSOR TAIT ON IMPACT. 



as shown above) that the friction is practically constant. Thus the motion of the block 



is represented by 



Mr = Mg ± F, 



the positive sign referring to upward motion. 



We have also, taking the angular velocity, <u, of the disc as uniform throughout the 

 short period of the experiment, 



eld = wdt. 

 Thus 



d 2 r ( , F\/ , ft _ 



w = \ 9 ± m)/- 2 - 2B > sa y; 



so that 



r = A + B0 2 , 



if we agree that 6 is to be measured in each case from the particular radius which is 

 vertical at the moment when the block is at one of its highest positions. 



If our assumptions were rigorously correct, the equations of those branches of the 

 curve which are traced during each successive rise of the block should differ from one 

 another solely in the values of the constant A. Similarly with those traced during 

 successive descents. The ascending and descending branches of the same free path 

 should differ solely by the change of value of B, according as the friction aids, or opposes, 

 the action of gravity. Also the two values of B should differ from their mean by a 

 smaller percentage the greater is the mass of the block. This, however, will be 

 necessarily true only if the friction be independent of the weight of the block. 



As a test of the closeness of our approximation, to be applied to the experimental 

 results below, it is clear that, if we call B the mean of the values of B for the parts of 

 the curve due to any one rebound, we have 



But, in the notation of the Tables as explained in the next section, we have 



o) = 27r/(6N/128). 



Taking the value of g as 32*2 when a foot is unit of length, it is 9814 to millimetres ; 

 and the two equations above give the following simple relation between B and N 



Q 



R — — N 2 



which is sufficiently approximate to be used as a test, the fraction being in defect by 

 about 0'14 per cent, only, say l/700th. 



Thus, in the first experiment of those given below for date 23/7/90, we have 



N = 21-25, 

 which gives as the calculated value 



B = 12316; or, -with l/700th added, = 12333. 



