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.

    18/104 (page 14)
    The first of these equations gives p = a. constant, for it evidently cannot be a function of t, since the effect of the motion vanishes at an infinite distance from the plane; and if we include this constant in 11, we shall have p = 0. Let V be the velocity of the plane itself, and suppose V=csmnt (9) Putting in the second of equations (8) V = Xi sin nt + cos nt, (10) we get nXi = /UL ,cPX, tl Xo — — ,d^X^ n'^d^x.. dos^ n dx* (11) The last of these equations gives (A Sin \/ 2 m (A sin \/—, X + B cos \/ a?) + e (C sin \/ x D cos \/ — x). 2m 2 m 2m' Since X^ must not become infinite when x = <x> , we must have C = 0, D = 0. Obtaining Xi from the first of equations (H), and substituting in (10), we get JUL- X / 71 / 7h V = € A sin (nt — \/ —+ Boos (nt - \/ — 2 m ^ M Now by the equations of conditions assumed in Art. 3, we must have v = V when a? = 0, whence -V^5^,^ . . / ^ \ V = ce ^ sin (nt — \/ —,x). 2m (12) To find the normal and tangential components of the pressure of the fluid on the plane, we must substitute the above value of v in the formulae (4), (5), and after differentiation put X = 0. Pi, Ts will then be the components of the pressure of the solid on the fluid, and therefore — Pj, — T3, those of the pressure of the fluid on the solid. We get Pi = 0, T3 = cp — (sin nt + co^ nt) = p \/ — I F+ - | . . (13) 2 2 \ n d/t j The force expressed by the first of these terms tends to diminish the amplitude of the oscillations of the plane. The force expressed by the second has the same effect as increasing the inertia of the plane. 8. The equation (12) shews that a given phase of vibration is propagated from the plane into the fluid with a velocity ^(‘Zfx.'n), while the amplitude of oscillation decreases in geometric progression as the distance from the plane increases in arithmetic. If we suppose the time of oscillation from rest to rest to be one second, w = vr; and if we suppose -y/ju = .II6 inch, which, as will presently be seen, is about its value in the case of air, we get for the velocity of propa- gation .2908 inch per second nearly. If we enquire the distance from the plane at which the amplitude of oscillation is reduced to one half, we have only to put \/ —, x = log^ 2, which 2m gives, on the same suppositions as before respecting numerical values, x = .06415 inch nearly.