Chap. 7] 



GRAVITATIONAL METHODS 



99 



in which for most practical applications it is suflEicient to use the angle for 

 its sine so that the "amplitude reduction formula" is 



r„ = r<,(i + ^+ ...y (7-m) 



For the physical pendulum of the mass M and the moment of inertia K, 



■Mgs sin 6 





Fig. 7-7. Physical 

 pendulum. 



where s is the distance of the center of gravity from the axis of rotation 

 (see Fig. 7-7). By comparison with equation (7-16a) it is seen that a phys- 

 ical pendulum, in which K/Ms = I = reduced pendu- 

 lum length, is isochronous with a mathematical pen- 

 dulum; its period T = 2ir -s/K/Mgs. 



The reversible pendulum (Fig. 7-8) is a physical 

 pendulum with two knife edges so placed that the 

 period of oscillation about either axis is the same. 

 Their distance is then equal to the length of the 

 equivalent mathematical pendulum. It is for this 

 reason that the reversible pendulum has been and is 

 still being used for the precise determination of abso- 

 lute gravity. The distance between knife edges may 

 be measured by means of a vertical comparator. 

 Determination of absolute gravity by means of the 

 reversible pendulum is a difficult procedure and requires a 

 number of corrections: (1) for the flexure of the support, (2) 

 for the effect of the surrounding air, (3) for the elastic tension 

 and bending of the pendulum, (4) for changes in temperature, 

 and (5) for the rate of the comparison chronometer. 



Inverted or near-astatic pendulums have the advantage of 

 smaller mass, greater periods, and greater sensitivity in period 

 to variations in gravity. The best-known representative is 

 the Lejay-Holweck pendulum. ^^ If an ordinary pendulum is 

 suspended from a spring instead of from a massless thread as 

 in Fig. 7-9a, the restoring force of gravity is added to that of 

 the spring. If its spring constant be designated by Co (see 

 pages 449 and 581), the equivalent spring constant of gravity 

 (force per unit elongation) would be mg sin 6 /a. Since sin 6 « 

 a/r, the resultant spring constant Cr = Co-\- mg/r. It follows further from 

 the equation for the elastic line that the equivalent axis of rotation is 



Fig. 7-8. 

 Reversible 

 pendulum. 



" Comptes Rendues, 186, 1827-1830 (1928); 188, 1089-1091 (1929); 190, 1387-1388 

 (1930); 192, 1116-1118 (1931); 193, 1399-1401 (1931); (1933). 



