108 



GRAVITATIONAL METHODS 



[Chap. 7 



curve (or the coincidence time interval n) these 60-second markers are 

 used as reference Hnes. Instead of the breaks themselves being counted, 

 the excursions on the upper or lower side of the record, such as d or a in 

 Fig. 7-14, may be used. If A^ is the angular phase lag of the pendulum 

 pair for a complete cycle 2ir, the number of oscillations required to complete 

 the 360° cycle is 27r/A^, and the period of the pendulum pair differs from 

 that of the chronometer in the proportion 27r/27r — A<p; thus, 



T = 



1 



= 0.5 + 



0.25 



n - 0.5 



(7-20) 



#**"*" 



2 2t — A<p 



where 2n = 27r/A^ = the coincidence interval. 



For absolute and relative determination 

 of gravity, various forms of pendulums have 

 been developed which are described in detail 

 by Swick.^* Two widely used forms are illus- 

 trated in Fig. 7-16. A is the Sterneck-type 

 quartermeter pendulum. The top part is a 

 stirrup holding a knife-edge made of agate 

 or quartz and two mirrors. 5 is a more re- 

 cent form known as the "rod" or '/mini- 

 mum" pendulum. In it the knife edge is 

 so placed that a change in its position has 

 a minimum effect on the period. In a 

 physical pendulum the moment of inertia, 

 K = IMs (see page 99), may be considered 

 as the sum of two moments, one with the 

 radius of gyration s about the knife edge and 

 the other with the radius of gyration r about 

 the center of gravity so that K = s M -{- 

 r^M and I = {r^ -\- s^)s. Hence, it follows 



Fig. 7-16. (A) Sterneck pen- by differentiation that 

 dulum; (B) Meisser bar pen- 



^'"^^"- dZ = ?r.i, + ^_ZJl.rf,. (7-21) 



For the least change of period T and therefore of reduced pendulum length 

 I with s, the factor of ds must be zero. This gives s = r and therefore 

 I — 2s. For "minimum" pendulums, (1) the reduced length must be 

 twice the distance of the center of gravity from the knife edge, s; (2) the 

 radius of gyration in reference to the center of gravity must be equal to the 

 distance s. It is not difficult to do this for circular rods, since r^ = L^/12 

 -|- R^/4:, where L is the geometric length and R the radius. 



" H. Swick, Modern Methods for Measuring the Intensity of Gravity, U.S. Coast 

 and Geodetic Survey, Serial No. 150. 



