Full text: Musschenbroek, Petrus: Physicae experimentales, et geometricae de magnete, tuborum capillarium vitreorumque speculorum attractione, magnitudine terrae, cohaerentia corporum firmorum dissertationes

CORPORUM FIRMORUM. circa Ellipſin, ad Cohærentiam Ellipſeos, in eadem ratione, ac
Cohærentia quadrati circa circulum, ad Cohærentiam circuli.

533. PROPOSITIO LIV.

Tab. XXV. fig. 9. Sit dimidiati Cylindri ſegmentum A B C D E,
cujus baſis rectangula A B C applicata parieti perpendiculari ad
horizontem: ſit parallelopipedum A B C E L M N, cujus baſis pla-
na A B C rectangula æqualis baſi Cylindrici ſegmenti, latus A B,
æquale A B, B C æquale B C, A E æquale radio D E in Cylindro,
erit momentum gravitatis in ſegmento Cylindrico ad momentum gra-
vitatis in parallelopipedo, uti duo ad tria.

Ponatur radius A D = r. peripheria circuli baſeos = p. Iatitudo
B C = a.

Erit area dimidii circuli A D B E A = {1/4} r p, quæ ducta in latitu-
dinem B C = a, dat ſoliditatem ſegmenti Cylindrici A B C A = {1/4} a p r. centrum vero gravitatis in ſemicirculo diſtat a centro D circuli quanti-
tate {8rr/3p}. in quam diſtantiam ducta ſoliditas, dat momentum
= {2/3} ar 3.

Soliditas parallelopipedi A B C E L M N eſt = 2 arr, hujus cen-
trum gravitatis eſt in medio, cujus directio tranſit per {1/2} A E = {1/2} r. adeoque erit momentum parallelopipedi = {1/2} r X 2 arr = ar 3 .

Eſt igitur momentum gravitatis in ſegmento cylindrico ad illud
in parallelopipedo: , {2/3} ar 3 . ar 3 : : 2,3.

Corol. 1. Si ergo ex latere B C parallelopipedi abſcindatur {1/3}pars,
per quam tranſeat ſegmentum parallelum ad ſuperficiem anteriorem
A B N M, erit momentum ex gravitate in parte reſidua parallelo-
pipedi æquale momento ſegmenti cylindrici = {2/3} ar 3 .

Corol. 2. Ut vero a parallelopipedo A B C L M N abſcindatur
pars, reliquumque habeat idem momentum gravitatis ac dimidia-
tus cylindrus, quæratur inter A M = r, ipſiuſque {2/3} partem media
proportionalis, quæ ſit = A O. tum per O K tranſeat ſegmentum
parallelum baſi A B C, habebit parallelopipedum A B C O K idem
gravitatis momentum, quod dimidiatus cylindrus; vocetur enim
A O, x, erit ſoliditas parallelopipedi A B C K O = 2 arx. ejuſque

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