Pendulum after the String has become slack. 281 



" potential ascent M lies above the point on the circle at a 

 height which is half the vertical height of this point above 

 the centre. The particle then describes a free parabola 

 having its directrix at this same level until the string 

 becomes taut at the intersection of parabola and circle. 

 This takes place at a point whose angular distance from the 

 highest point of the circle is three times the angular distance, 

 measured in the opposite direction, of the point at which the 

 thread became slack. It is easy to prove this analytically, 

 but it is most quickly obtained as a consequence of the well- 

 known geometrical theorem that the chords which join, in 

 pairs, the four intersections of a circle and a conic are 

 equally inclined to an axis of the conic. In the present case 

 the parabola and the circle have the same curvature at their 

 point of contact, and so three of the intersections coincide at 

 this point. The common tangent is one of the two chords, 

 and the other is the line joining the point of slackening to 

 the point of tightening of the thread. These are equally 

 inclined to the vertical, and the given result follows 

 immediately. 



If the thread be supposed inextensible the component 

 velocity in its direction is destroyed by the jerk, and the 

 bob resumes its movement in the circle with the tangential 

 component of its previous velocity. 



If a is the angular distance from the highest point at 

 which the thread slackens, the tangential velocity when it 

 tightens again comes out to be 



\f ga cos « (8 cos 4 a — 12 cos 2 a. + 3), 



where a is the length of the thread. This vanishes when 

 cos 2 a = i(3— V3) or « = 55° 44'. Accordingly if the 

 thread slackens at this point the bob will be brought 

 instantaneously to rest when the thread makes with the 

 downward vertical the angle it— 3a = 12° 48', and will then 

 oscillate with this amplitude. 



In general, the level of no velocity after the jerk is at a 

 height above the centre of the circle given by 



\a cos a (3 — 64 sin 6 « cos 2 a). 



This is shown plotted against a in the middle curve on the 

 diagram. The values of a run backward, so that the origin 

 corresponds to the end of the horizontal diameter of the circle. 

 It is convenient to start from this position in discussing the 

 motion. 



The upper curve gives the level of no velocity before the 



