Logarithmic Decrement 



121 



WHY THE LOGARITHM IN LOGARITHMIC DECREMENT? 



R. R. Ramsey, Indiana University. 



When using a ballistic galvanometer it is usual to observe the first de- 

 flection and the second deflection and multiply the first deflection by the factor 

 1 + H>^ where X is the logarithm of the ratio of the first deflection di to second 

 deflection do, in order to get the deflection which should have been obtained 

 if the galvanometer were not damped. 



If the galvanometer is not damped we have simple harmonic motion. 

 Simple harmonic motion may be considered to be uniform circular motion 

 projected upon a diameter of the circle. In like manner the motion of the 

 galvanometer when damped may be considered to be the projection of the 

 motion of a body moving with imiform angular motion but spiraling in towards 

 the center upon a curve known as the logarithmic spiral. 



Fig. 1. Projection of logarithmic spiral. 



Figure 1 represents a circular of radius do and inside of which is the spiral 

 whose equation is d = doe '^^ ; where d is the distance from the center at 

 any point, do is the initial distance at the point, a — in this case do is the radius 

 of the circle also, ti is the angle measured from a and a is a constant. 



The first, second, etc., deflections are the horizontal distances from the 

 center o, when d = tc/2, 3 x/2, etc. 

 Then since d = doe ~ "'^ 



di = doe -W2 

 d, = doe -«3x/2 

 da = doe - «5x/2 



"Proc. Ind. Acad. Sci., vol. 33, 1923 (1924)." 



