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, 6 = when 6= a, and the equation of energy of that 

 Article can be written 



. - 



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 ^r defined by the equation 



OL . .6 



sm - sm \fr = sin - , 



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

 increases from to JTT ; as 6 diminishes from a to 0, ty increases 

 from JTT to TT ; as 6 diminishes from to a, ty increases from TT 

 to |TT; and as 6 increases from a to 0, ^ increases from f?r to 

 2?r. With these conventions there is one value of i|r correspond 

 ing to every instant in a complete period. 

 Now we have 



cos - = -^ sin ^ cos ^, 



sin 2 - sin 2 = sin 2 cos 2 \r, 



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 



n 



Vgh 



