Bernoulli, Daniel - Hydrodynamica, sive De viribus et motibus fluidorum commentarii

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5844HYDRODYNAMICÆ

De Effluxu Aquarum ex Cylindris verticaliter po-
ſitis, qui in alios tubos ſtrictiores pariter
verticales deſinunt.

§. 21.

COnſtat experientia, inter duos Cylindros omnino æquales ſimiliterque
poſitos, quorum alterius foramini tubus ſtrictior reſpondeat, hunc
citius depleri, qui tubum appenſum habet, & quidem eo citius, quo
magis tubus à loco inſertionis verſus extremitatem amplitudine creſcit, quæ
pluribus expoſuit D. s’Graveſande in Phyſ. Elem. Math. lib. 2. cap. 8. Totam rem
ſequenti Problemate comprehendemus.

Problema.

§. 22. Fuerit vas cylindricum A E H B (Fig. 18.) verticaliter poſi-
11Fig 18. tum perforatum in F G, quo lumine communicet cum tubo conico F M N G,
per cujus demum orificium M N aquæ effiuant. Quæritur velocitas ſuperfi-
ciei aqueæ C D, poſtquam à quiete deſcendit per A C vel B D.

Solutio.

Sit altitudo aquæ ſupra M N initialis, nempe N G + H B = a, altitu-
do ſuperficiei aqueæ in ſitu C D ſupra M N, id eſt, N G + H D = x; lon-
gitudo tubi annexi ſeu N G = b; amplitudo orificii M N = n; amplitudo
orificii F G = g, amplitudo Cylindri ſuperioris = m; ſit velocitas ſuperficiei
aqueæ in C D talis quæ debeatur altitudini v, erit in æquatione generali §. 8.
y = m & N = m (x - b) + {bmm/√gn}, quæ ſubſtitutiones inſtituto calculo con-
formes eſſe patebunt cum §. 6. reliquæ autem poſitiones eædem ſunt quæ an-
te. Abit igitur æquatio paragraphi 8 in hanc
m(x - b)dv + {bmm/√gn}dv - {m3vdx/nn} + mvdx = - mxdx
quæ porro diviſa per m factoque x - b + {mb/√gn} = z,

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