180 MOTION UNDER CONSTRAINTS AND RESISTANCES. [CHAP. X. 



9. Find the greatest angle through which a person can oscillate on a 

 swing, the ropes of which can support a tension equal to twice the person's 

 weight. 



*191. Finite Oscillation. More generally, suppose the 

 pendulum to start from rest in a position in which the radius 

 makes an angle a with the vertical. Then, in the notation of 

 Article 188, $ = when 6 a, and the equation of energy of that 

 Article can be written 



^l& = g (cos 6 cos a), 



or i^ 2== f 



showing that the pendulum oscillates between two positions in 

 which it is inclined to the vertical at an angle a on the right and 

 left sides of the vertical. 



To express the position of the pendulum in terms of the time t, 

 since it was in the equilibrium position, we introduce a new 

 variable ^ denned by the equation 



a . 6 



sin ^ sin y = sin = , 



' 2i 



with the further conditions that as 6 increases from to a, ^ 

 increases from to \TT ; as diminishes from a to 0, ty increases 

 from |TT to TT ; as 6 diminishes from to a, ^ increases from TT 

 to ITT; and as increases from a to 0, ty increases from -| TT to 

 2?r. With these conventions there is one value of ^r correspond- 

 ing to every instant in a complete period. 

 Now we have 



i4 J 



\V COS - = i/r sm 2 COS 1/r, 



a 6 a. 



sin 2 sin 2 -= = sm 2 -= cos 2 y, 



Hence the time t from the instant when the particle was 

 passing through the lowest point in the direction in which 6 

 increases is given by the equation 



jl_ { 

 "V^Jc 



Y (l- 



