CONTRIBUTIONS FROM THE JEFFERSON PHYSICAL LABORATORY, 



HARVARD UNIVERSITY. 



THE DAMPING OF THE OSCILLATIONS OF SWINGING 

 BODIES BY THE RESISTANCE OF THE AIR. 



By B. Osgood Peirce. 



Presented March 11, 1908. Received June 29, 1908. 



When a body, free to turn about a fixed axis, like a horizontal pen- 

 dulum, a suspended magnet, or the coil of a d' Arsonval galvanometer, 

 is disturbed from a position of equilibrium, and is then allowed to 

 swing under the action of a righting moment the intensity of which 

 is proportional to the angular deviation of the body from the position 

 of rest which it originally had, the damping effect of the resistance 

 which the air offers to the motion is sooner or later made evident by 

 a reduction in the amplitude of the swings. In many cases the phe- 

 nomena can be quantitatively explained, with an approximation quite 

 good enough for every practical purpose, if one assumes that the re- 

 sisting couple has a moment equal at every instant to the product of 

 a constant of the apparatus and the angular velocity which the body 

 then has ; and more than seventy years ago Gauss and W. Weber gave 

 an exhaustive mathematical treatment, 1 based upon this hypothesis, 

 of the behavior of such swinging magnets as they employed in their 

 magnetic measurements at Gottingen. It appeared from their analysis, 

 which in simplified form is given in most modern treatises on Physics, 

 that if the resistance follows the law stated above, the ratio of any two 

 successive elongations of the magnet must have a constant value ; and 

 they used the natural logarithm (X) of this ratio, under the name 

 of the "logarithmic decrement" of the motion, in many of their 

 equations. 



The resistance which air, under given conditions of temperature, 

 pressure, and confinement, offers to a body of given form and dimen- 

 sions, moving through it at a uniform velocity, v, has been studied by 



1 Gauss, Resultate des Magnetischen Vereins, 1837. W. Weber, Resul- 

 tate des Magnetischen Vereins, 1837, 1838; Maassbestimmungen, 2; 

 Math.-phys. Abhandlungen der K. Sachs. Gesellschaft, 1852. Du Bois-Rey- 

 mond, Monatsberichte der Berl. Akad., 1869, 1870. 



