98 



GRAVITATIONAL METHODS 



[Chap. 7 



sumed by friction or damping. The energy for the maximum amplitude, 

 a, IS m-g -1(1 — cos a), and is equal to the sum of the potential energy, 

 m-gl{l — cos 6), and the kinetic energy, \'m-t-{dd/dt)^, for the position d. 

 Hence, 



= ./x ^ 



y 2a;2 



dt 



(7-16c) 



\/cos d — cos a 



The period Ta is twice the time required^ for the pendulum to swing from 

 Q = a to d = —a. By substitution of 1 — 2 sin^ 6/2 for cos d and 

 1 — 2 sin'^ a/2 for cos a, 



dd 



(7-16d) 



r« = 



sin 



2 _ _ 



sm 



2^ 



By introducing the auxiliary angle ^, so that sin 

 6/2 = sin ^ sin a/2 and 



d0 = 



2 sin - cos ^ d^ 



/ 



Fig. 7-6. Mathematical 

 pendulum. 



the period 



T = 



1 — sin^ ^ sin^ v^ 



# 



1 — sin^ ^ sin^ ^ 



The elliptic integral has the form 



# 



/ , Vl - p^sin^t//' 

 whose solution, (see B. 0. Pierce, Table of Integrals, No. 524) is 



so that the period 



_ 2ir /- , 1 . 2 « , 9 . 4 a , \ 



If the period for small amplitudes is To , 



r.= r.(^l+-sm2 + g4sm -+...J, 



(7-16e) 



(7-16/) 



(7-16^) 



" Partly after L. Page, Theoretical Physics, Van Nostrand (1928). 



