812 
MR. H. TOMLINSON ON THE INFLUENCE OF STRESS AND 
Formula for determining the Loss of Energy experienced hy a Metal Wire vibrating 
under the Influence of Torsional Elasticity and encountering a Resistance pro¬ 
portional to the Velocity. 
The diminution of amplitude of a wire vibrating as in the present experiments is 
due to three causes :— 
1. Energy imparted to the support to which the upper extremity of the wire is 
attached, or to the parts of the vibrator which may be capable of independent motion.* 
2. Energy imparted to the air surrounding the wire and vibrator. 
3. Internal molecular friction. 
Of these causes, 1 may be dismissed at once as not affecting the results within the 
limits of errors of observation, and a means of eliminating the effect produced by 
the resistance of the air will be shown presently. I propose, therefore, in dealing 
with the problem before us, to confine myself to the consideration of Cause 3, on the 
assumption that the internal friction varies as the first power of the velocity. 
Let F be the moment of the couple necessary to twist the wire through one radian ; 
2 k the resistance due to internal friction, when the wire is twisting or untwisting with 
unit angular velocity; r the period of vibration from rest to rest: and M the moment 
of inertia: then it can readily be shown that very approximately 
T = ,r Vf( 1+ 2^}.W 
and that the proportionate diminution of amplitude is 
( 2 ) 
From (2) it follows that the logarithmic decrement should be independent of the 
amplitude of vibration, and should vary inversely as the period of vibration. 
Discussion of Thomson’s Views concerning the interned Friction of Metals. 
According to the mathematical formula just given, the logarithmic decrement should, 
with the same vibration-period, be a constant for all amplitudes if the friction between 
the molecules of a metal resemble fluid friction. This we have seen is the case.t 
When I turned, however, to examine the effect of changing the moment of inertia of 
/i‘7T" 
tF* 
* If, for example, the cylinders attached to the bar of the vibrator be capable of a motion independent 
of the bar, as they would be if suspended by fine wires or threads. 
t Except, of course, in such cases as that of Experiment II., where the synchronism between 
torsional and pendulous vibration-periods causes the logarithmic decrement to vary; or, rather, prevents 
there being any such thing as a logarithmic decrement. 
