i 809 ] 



LXXIII. The Motion of a Simple Pendulum after the String 

 has become Slack. By Arthur Taber Jones, Ph.D., 

 Associate Professor of Physics at Smith College, U.S.A. * 



Introduction. — Under the above title Professor W. B. 

 Morton recently published f a very interesting note. The 

 pendulum is supposed to swing in a plane, the string to be 

 inextensible, and the velocity of the bob to be sufficient to 

 carry it higher than the point of support, but not sufficient 

 to bring it to the top of its circular path. At some point, 

 higher than the centre of the circle, the tension of the string- 

 vanishes and the path of the bob becomes a parabola. At 

 the point where this parabola intersects the circle, the string 

 tightens again and jerks the bob out of its parabolic path. 

 In ideal cases this jerk may be thought of as perfectly 

 elastic or as perfectly inelastic. If it is perfectly elastic it 

 reverses the radial component of the bob's velocity, and if 

 it is perfectly inelastic it destroys this radial component. 

 In the case which Professor Morton has discussed the jerk is 

 treated as perfectly inelastic. Throughout the present note 

 the jerk is treated as perfectly elastic J . With a real string the 

 jerk is far from being perfectly elastic or perfectly inelastic, 

 so that the case of a real pendulum is an intermediate one. 



The general case. — If the jerk is perfectly elastic, each 

 path consists, in general, of a circular arc followed by an 

 infinite series of parabolic arcs. For the first parabola 

 Professor Morton points out .that the level of no velocity is 

 given by 



Z = |rcosa, (1) 



where I stands for the distance from the centre of the circle 

 up to the level of no velocity, r for the radius of the circle, 

 and a for the angle which the vertical diameter of the circle 

 makes with the radius to the point where the string first 

 slackens. On the present hypothesis no energy is lost in the 

 jerks, so that (1) gives the level of no velocity throughout 

 an entire path — that is, for every one of an infinite series of 

 parabolas. 



If we take the origin at the centre of the circle, and 



* Communicated by the Author. 



t Phil. Mag-. (6) xxxvii. p. 280 (1919). I have checked Professor 

 Morton's results and obtained the same expressions that he has. There 

 is a misprint at the bottom of p. 282, where the equations should read : 

 f /'(a 2 ) = I cos cr-L, < /(a 3 ) = | cos a 2 . 



j This problem may, of course, just as well be thought of as having 

 to do with the motion of a particle which slides and bounds in a single 

 plane inside of a frictionless spherical cavity, the impacts being- perfectly 

 elastic. 



