On the effect of the internal friction of fluids on the motion of pendulums / by G.G. Stokes.

  • Stokes, George Gabriel, 1819-1903.
Date:
1851
Catalogue details

Licence: Public Domain Mark

Credit: On the effect of the internal friction of fluids on the motion of pendulums / by G.G. Stokes. Source: Wellcome Collection.

Provider: This material has been provided by The Royal College of Surgeons of England. The original may be consulted at The Royal College of Surgeons of England.

    100/104 (page 96)
    Now treating At as a slowly varying parameter, we get from a formula given by Mr Airy, and obtained by the method of the variation of parameters, dAt G . —— = — sin w A dt I (191) where I denotes the moment of inertia. In the expression for G we may replace At_t^ under the integral sign by Ax outside it, because Ai is supposed to vary so slowly that At-t^ does not much differ from At while ty is small enough to render the integral of importance. Making this simplification and substituting in (191) we get 'At * A dt\ -—=-cs\nnt / cosw (« - #i) —- , . . . . (192) tdt Jq .y/ dAt A where c = y^{7rfx) . pa*nl~^. If then be the initial and A the final value of At, we get from (192) 1 Aq r* f • r* ^ 1 log—=c / ism nt cos n {t - ti)dt (193) *^o I •'0 V ^iJ Let now Aq + A Ag be what A^ would become if, while the final arc A and the whole time t remained the same, the motion had been going on for an indefinite time before the epoch from which t is measured, in which case the superior limit in the integral involved in the expression for G would have been 00 in place of t. Then log = c f jsinw# / cos n {t - dt, . . (194) A *'o I *'0 'v whence by subtracting, member from member, equation (193) from equation (194), we get . Ag + AAg log 1 = cj ^cosn(t - ty)~~^dt, which becomes after integration by parts , Ag+ AAg c (/ TT , , loe = — < V/ 2a/^. cos nt — cos 2nt / coswr ° Ag 4W \ 2« ^ Jt X r” • d# + (2nt — sm2nt) / sin —— Jt t dt ~^t (195) Now t is supposed to be very large: in Coulomb’s experiments in fact 10 oscillations were observed, so that nt = 10tt. But when t is at all large the two integrals I “ dt cos n t —j-, -y' i sin nt dt can be expressed under the forms — P sin nt + Q cos nt, P cos nt + Q sin nt, where P = n~^ t~^ —1 .3.2~^n~^t~\ + ..., Q = 1.2~^n~‘^t ® — 1.3.5.2 ^n~*t 5 + ...,