260 MOTION UNDER A VARIABLE FORCE 



Writing the equation in the form 



dv 



v-r = - 

 ds \a 



and integrating with respect to s, we obtain 



(95) 



Clearly the constant C must be positive, and the velocity will 

 vanish as soon as s reaches a value such that 



Let us denote the two values of s which satisfy this equation by 

 . s , then the motion of the bob is clearly confined within two 

 points at distances s from the point on opposite sides. Thus 

 we may call s the amplitude of the swing. 



Replacing C by ( -js 2 , equation (95) becomes 



* 2 =!(*o 2 -* 2 ), (96) 



so that 



and the integral of this equation is 



t =, r ds 



J AJ^o- 



where e is a constant of integration. 

 This equation gives 



so that s = s cos 



