282 Prof. W. B. Morton on the Motion of a simple 



jerk, the lowest curve the level at which the thread tightens. 

 These are the sine-curves y = §a cos a and ;/ = a cos 3« 

 respectivel}". The vertical distance between the top curve 

 and the middle one represents the energy destroyed by the 

 jerk. The distance of the middle curve above the bottom of 

 the diagram is the total energy after the jerk, and it is 

 divided by the lowest curve into two parts : the upper part, 

 between the two curves, is the kinetic energy, and the lower 

 part the potential energy measured from the lowest position 

 of the bob. It will be seen that the carves touch for 

 a = 55° 44', i. e. the bob is then stopped dead as already 

 explained. 



By use of these graphs it is easy to follow the complete 

 course of the motion of the bob when the thread slackens for 

 the first time at a given position on the circle. If the level 

 of no velocity after the jerk is below the centre the pendulum 

 will oscillate below this level. If it is above the centre the 

 thread will slacken again. To find where this takes place 

 run a horizontal line from the point on the middle curve to 

 meet the upper curve. Underneath the point of intersection 

 is found the value of a. at which the thread becomes slack 

 for the second time. 



When this process is repeated it is found that the ultimate 

 motion alternates between two types : — 



(1) Asymptotic approach to oscillation between the ends 

 of the horizontal diameter. This will occur if the thread 

 slackens anywhere on an arc of 13° 39' above an end of this 

 diameter (from a=:90° to a = 76° 21'). The curve giving 

 the level of zero velocity after the tightening of the thread 

 is then above the axis, and if we apply the construction to 

 find successive points of slackening we are led by a series of 

 zig-zags into the corner, at u = 90°, between the upper and 

 middle curves. The same terminal state of motion is reached 

 if the 2nd, 3rd, 4th .... slackening falls within this region, 

 which will be the case if the first slackening falls in the 

 regions shown on the diagram by a thickening of the axis of «. 

 The boundaries of these segments are found by construction 

 as indicated by the lines drawn vertically and horizontally, 

 beginning with the two points a = 76° 21' and 41° 21', 

 where the middle curve crosses the axis. They can be cal- 

 culated by solving by trial the series of equations 



/( ai ) = 0, /(a 2 )==|a l5 /(« 3 )=:§a 2 , and so on, 



where /(a) = i cos a (3 64 sin 6 a cos 2 a). 



It is thus found that the motion runs into the asymptotic 



