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.

    21/104 (page 17)
    the velocity in the direction of “Zer, w the angle between the line -zar and the plane of xy. Then p, u, and q will be functions of x, and and we shall have whence V = q cos £t», w = sin y =■ ny cos ai, % = ny sin w, •zzr® = y® 4- is®, ^ = tan”^ -. y We have now to substitute in equations (2) and (3), and we are at liberty to put o) = 0 after differentiation. We get sin (!) d whence we obtain d d — = cos OJ — dy disy ny di — = when ftj = 0, dy d-sr d . d cos w d — = sinw— + — , a% any ny acD d = -— when w = 0, dny 1 ^ u when 0) = 0, ny do) <P Id 1 d^ — = when o) = 0, ny dny tjt doi dp du dq ~r +1— 1 du\ du w d-sy) ^ dt ’ . . . (16) 1 dq q \ dq ny dw -bP) ^ dt’ ' . . . (17) + - = 0 . . . (18) Eliminating p from (l6) and (l?)? and putting for its equivalent //p, we get ! d f d^ (P ld\ , d I d^ d® id d /du dq\ ^ dny \doP dny^ ny d-srJ ^ ^ dx \da?® ^ dny^ tst dny ny^J ^ dt \dny dx) ’ d^ or d / (P dx \( 1 d f d^ d' id 1 1 d\ (du dq\ Vda?® dny ny dw ny^ a dt) \dny dx) By virtue of (18), ny (udny - qdx) is an exact differential. Let then “zir (udny — qdx) = d\J/ Expressing u and q in terms of \j/, we get du dq 1 / P P ld\ d-zsT dx ■ztT dny^ rsy dwj Substituting in (19)> and operating separately on the factor nr~^, we obtain /d® d® Id IdWd^ P ^ ^ \ ^ _ Q \dx^ d^ny w din p! dt) \dx dny^ Tsy dny) (19) (20) (20') 3