100 



GRAVITATIONAL METHODS 



[Chap. 7 



Fig. 7-9a. Sus- 

 pended elastic 

 pendulum. 



Fig. 7-96. In- 

 verted elastic 

 pendulum. 



Fig. 7-10. Le- 

 jay-Holweck pen- 

 dulum. 



located approximately one-third of the spring length 

 from the point of suspension. Substituting, there- 

 fore, fZ for r and SEJ/l for the spring constant Co 

 (where E is Young's modulus of elasticity and J the 

 moment of inertia of the spring section), the resulting 

 spring constant Cr = 3EJ/1 + 3mg/2l, so that by sub- 

 stitution into CO = -s/c/m : 



-^, 



SEJ 



3g 

 21 



(7-17a) 



In the inverted pendulum the action of gravity tends 

 to drive the mass away from the rest position instead of 

 toward it (Fig. 7-96) ; hence, 



'/ 



3EJ 



mP 



21 



or 



} (7-176) 



= 2X|/; 



2ml^ 



QEJ - •imgP 



The change of period with gravity is given by dT = dg 

 Tol/{2EJ - mgf). T becomes infinite when m = 2EJ/fg. 

 In a derivation not involving the approximations made 

 here, the factor is 7r"/4 instead of 2.^" Numerical eval- 

 uation of eq. (7-176) shows that in order to obtain any 

 advantage in sensitivity, the mass has to be made so 

 large that the buckling strength of the spring is 

 approached. This can be avoided by using a long bar 

 and a short spring; in the Lejay-Holweck pendulum the 

 length I is several times smaller than the distance L 

 (see Fig. 7-10). With K as the moment of inertia of 

 the pendulum mass, the period and its change with 

 gravity 



T = 2ir 



l/o7 



K 



mgV 



dT = 



T 



dg^ 



2 Co — mgh' 



(7-17c) 



With the dimensions used in the Lejay-Holweck 

 pendulum, a change in period of 1 • 10~ seconds corre- 



ct* A. Graf, Zeit. Geophys., 10(2), 76 (1934). 



