April 14, 192 1] 



NATURE 



209 



attainable in these days. The reason is to be 

 sought in the fact that the conditions of constraint 

 under which the impact experiments are made are 

 essentially different from those under which 

 a golf ball is compressed and distorted as it 

 is propelled freely in its flight. Everyone knows 

 that the high resilience of the rubber-cored ball 

 is derived from the fine rubber thread which is 

 wound on under considerable tension. Before the 

 outer covering is put on, these balls, when dropped 

 from a height of 6 ft. or 7 ft., rebound from 

 stone or metal to a height which indicates that 

 the coefficient of restitution exceeds o-8. When 

 we consider the manner in which this complex of 

 tightly wound rubber resists any sudden distor- 

 tion produced by a short-lived blow, we shall prob- 

 ably be prepared to admit that such an elastic 

 complex will resist compression more powerfully 

 than an equal sized ball of vulcanised india-rubber, 

 which Tait found to have a coefficient of restitu- 

 tion greater than 08. Any impulse brought to 

 bear upon one part of the rubber-wound ball will 

 produce in every strand of the rubber thread an 

 immediate tightening with corresponding resist- 

 ance to change of shape. 



Let us suppose, then, that, under these con- 

 ditions, the coefficient of restitution approaches 

 the value unity, say 0-95. If the old gutty with 

 coefficient of restitution 066 was propelled with 

 an initial velocity of 300 ft. /sec. , then this ball, 

 with coefficient of restitution 095, will be pro- 

 jected with initial speed of 356 ft. /sec. This by 

 itself will not account for an increase of 70 or 

 So yards in the length of drive, for, as pointed 

 out by Tait, a greater initial speed means a 

 ^Teater air resistance ; and (other things being the 

 same) to add 83 yards to the length of a drive 

 means double the velocity at start. But here, 

 again, we may invoke the influence of the under- 

 spin. As already stated, there is for every velo- 

 city of projection a definite value of underspin 

 which will enable a given ball to travel its 

 farthest range. Since the upward force produced 

 by the combined action of the linear velocity and 

 spin depends on both these factors, an increased 

 \elocity of projection will have to be associated 

 with an increased rate of spin if its greatest range 

 is to be attained. The problem is one which would 

 well repay working out in detail. 



If great length of drive is a desideratum in the 

 game of golf, then undoubtedly the " floater " 

 must give way to the heavy ball. This is a simple 

 illustration of the well-known law of atmospheric 

 resistance, the effect of which upon a sphere pass- 

 ing through the air is directly proportional to the 

 surface, and inversely proportional to the mass. 

 The accurate driver finds by experience that a 

 heavy small ball travels farthest through the air. 

 For example, if we make a floater of density 

 unity and of the maximum weight, its diameter 

 will be 1-75 in. The retarding effect of the re- 

 sistance of the air on this floater will be 17 per 

 cent, greater than the retardation experienced by 

 the new standard ball of minimum size and maxi- 

 mum weight. Again, if we make a floater of the 

 NO. 2685, VOL. 107] 



minimum size, its weight will be only 1-28 oz. , 

 and it will experience a retardation due to atmo- 

 spheric resistance which will be nearly 27 per cent, 

 greater than that experienced by the standard 

 "minimax," to use a word introduced long 

 ago by Kelvin in a different connection. The 

 "minimax" itself experiences slightly less atmo- 

 spheric resistance than most of the balls men- 

 tioned above, being excelled in this respect only 

 by Dunlop 31, Spalding Midget, small Avon de 

 Luxe, and Silver King ; but the difference never 

 reaches 2 per cent. It is therefore not surprising 

 that long driving is also attainable with the 

 standard "minimax" ball. 



A reference has been made to the radius of 

 gyration of a golf ball as a factor influencing the 

 amount of spin communicated to the ball. The 

 square of the radius of gyration of a uniform 

 sphere is fr^, where r is the radius of the sphere. 

 By means of oscillatory experiments, in which the 

 golf ball was supported by a ring-shaped disc 

 hung by a tri-filar suspension from three fixed 

 points, the moments of inertia and radii of gyra- 

 tion of all the golf balls used were determined to 

 an accuracy of about i per cent. The moments of 

 inertia expressed in grams and centimetres varied 

 from 86 for the Large Heavy Avon to 66 for the 

 Standard Clincher Cross, and yet the mass of the 

 latter was slightly the greater, being 454 grams 

 (i-6o oz.), as compared with 446 (1-57 oz.). This 

 great difference in the moments of inertia depends 

 on the distribution of matter within the ball. The 

 value of fe2 for the larger balls was practically the 

 same as the value 'ir^ for the uniform sphere of 

 equal size ; but in the case of the small balls fe- 

 was markedly less than f r^, being in some cases 

 as much as 8 per cent, smaller. The reason is 

 that the small balls have a very dense core. It is 

 obvious that with the larger moment of inertia a 

 greater moment of impulse must be given to 

 obtain the same spin. But this is automatically 

 effected, since with the same club the larger ball 

 is struck along a lower line relative to the centre 

 of mass, so that the moment of the impulse is 

 of necessity greater. During the flight of the ball 

 the larger moment of inertia will enable the ball 

 to conserve its spin the better, which will prob- 

 ably have a beneficial effect on the range or carry. 



It appears, then, that the length of drive attain- 

 able depends on several factors, and of these the 

 most effective are the resilience of the ball and 

 the underspin given at the instant of impact. To 

 drive a long ball is one of the delights of golf, 

 and the ball which travels farthest will be the 

 favourite. By almost all young and vigorous 

 players the floater, because of its lightness, is re- 

 garded unfavourably. It lacks, comparatively 

 speaking, steadiness in the air and accuracy on the 

 greens, and cannot possibly be driven so far. It 

 is little wonder that the heavy ball has ousted it In 

 all serious plav. 



It is not the purpose of this article to touch on 

 the question of standardisation of the golf ball. 

 Its aim is to discuss the physical principles which 

 govern the flight of the ball through the air. But 



