August 8, 1919] 



SCIENCE 



139 



196,000 — 82,000 = 114,000, the new region 

 representing tlie increase in frequency num- 

 ber (oscillations per centimeter) of 312,000 — 

 196,000 = 116,000. 



E. A. MiLLIKAN, 



E. A. Sawyer 

 Eterson Physical Labokatoet, 

 University of Chicago, 

 Chicago, III. 



THE PROBLEM OF THE BOY IN THE SWING 



In the current issue of Science (p. 20), 

 Professor A. T. Jones has given an excellent 

 account of just hov7 a boy works up in a 

 swing. To solve a problem in physics qualita- 

 tively and experimentally and at the same 

 time to keep the explanation clear and correct 

 as Professor Jones has done is often much 

 more difficult than to explain the same phe- 

 nomenon quantitatively. ISTevertheless, his last 

 paragraph, dealing with the energy relations, 

 aroused my curiosity to discover just what the 

 equation is which connects the work done by 

 the boy's muscles with the increased rotational 

 energy of the swing. 



What here follows is practically as old as 

 Huygens and is well knovm, but may interest 

 those who have read the note referred to. 



If the distance of the center of gravity of 

 the boy from the limb about which the swing 

 rotates be denoted by r, his mass by m; and 

 the angular speed of the swing by w, then his 

 angular momentum will be mr-w. Suppose 

 now that the boy who has hitherto been stand- 

 ing up in the swing proceeds to sit down upon 

 his heels; then if his angular speed is to be 

 maintained equal to that of a rigid pendulum 

 (isochronous with the swing loaded with the 

 standing boy and vibrating through the same 

 amplitude) a torqe, L, must be introduced 

 whose value, at each instant, is 



L = '^-^ = 2mrr.. 

 at 



Or, if no such external torque be applied, 

 then the boy's motion will be retarded, at each 

 instant, by just this torque. 



The tangential force which opposes the mo- 

 tion, as the boy moves away from the axis, will 

 evidently be 



F = - = 2mru, 



T 



a quantity which becomes zero whenever either 

 the radial speed, r, or the angular speed, w 

 vanishes. Except for very small angles of 

 deviation, 6, this retarding force will be but 

 a small fraction of the tangential component 

 of the weight, mg sin 6, which is urging the 

 loaded swing to its lowest point. 



When the hoy rises to a standing position, 

 the sign of r changes and his motion, instead 

 of being retarded, is accelerated. Here is 

 where the kinetic energy of the pendulimi is 

 increased; and the amount of it. if ds be an 

 element of length of the arc, will be 



Fds = 2mroids. 



But since <,> is much greater near the middle 

 than near the end of the vibration, the boy 

 will expend more energy in lifting himself at 

 the bottom of the swing than he will gain in 

 seating himself at the end of the swing; this 

 quite aside from the fact that, at the lowest 

 point, he works against the whole of gravity 

 while at the maximum elongation only the 

 radial component of his weight is effective. 



To perform the actual integration of the 

 above expression one would have to know — or 

 assume — the rate at which the boy seats him- 

 self, i. e., one would have to know r as a func- 

 tion of s. 



The phenomenon is, of course, not neces- 

 sarily associated with gravity. The same 

 description would hold for a mass in radial 

 motion along the spoke of an oscillating hori- 

 zontal wheel — say, the balance wheel of a 

 watch. 



For the student of dynamics, the essential 

 interest of the problem appears to lie in the 

 general fact that, although a central force 

 does not alter the angular momentum of a 

 body about a perpendicular axis through the 

 center, such a force will, imless balanced, 

 affect the kinetic energy of the body. Any 

 one who wishes to imderstand this fact will 

 try for himself the simple pendulum experi- 

 ment recommended by Professor Jones, no 

 matter how vivid his boyhood recollection of 



