272 EXPLORATION GEOPHYSICS 



equivalent to a simple pendulum of length k where h = — 77-. Hence, a com- 

 pound pendulum may be used to determine the absolute value of g provided 

 the equivalent length h can be evaluated. 



To determine the equivalent length k, use is made of the fact that to 

 every point of suspension 5 there corresponds another point S\ known 

 as the center of oscillation, which has the property that the period of vibra- 

 tion about it is the same as the period of vibration about S. The proof 

 that there is such a center of oscillation is readily shown by using a well 

 known relation between the moment of inertia /c about an axis through 

 the center of mass C and the moment of inertia /g about a parallel axis 

 through S. That is, 



7g = /g + mhi^ or /« = mk^ + mhi^ 



where k is the radius of gyration about the center of mass. If one sub- 

 stitutes the latter value for /s into the expression for the period one obtains 



_, ^ Jmk^±mh^ T - 9 \//' ^' 4- ^ V 

 ^^ = 2.^ ^j^^g or rx-2.\(^^ +h,)- 



Hence, the length li of the equivalent simple pendulum may be written in 

 the form 



In the same manner it may be shown that the period of vibration about 

 a parallel axis through another point such as S' on a line through Sc 

 extended is 



T. = 2Wg+/.= )i 



where /12 is the distance between parallel axes through S' and C. The 

 length I2 of the equivalent simple pendulum for this case is 



/2 = T + ^2 



^2 



For /i = I2 the periods of vibration about the two axes S and 5' will be 

 the same. This condition may be written in the form 



1= li = l2 = -, — ^ hi = -j — 1- /12 

 hi «2 



On elimination of k, this condition becomes 



1= hi + h2 



which is the relation sought. 



