We address the location of the metacenter of a floating body such as a ship. Previous studies of in relation to the stability of a ship have mainly used geometrical approaches and were limited to near equilibrium. We develop a quantitative approach to the location of for a general shape of the cross-section of a floating body in a rolling/pitching motion and for an arbitrary heel angle. We show that different definitions of refer to one and the same special point of the floating body. We discuss the relation between the height of with respect to the line of flotation and the contribution of the buoyancy force to ship stability. We provide expressions and graphs of the buoyancy, flotation, and metacentric curves for some simple shapes of floating bodies.
REFERENCES
1.
The Works of Archimedes
, edited by T. L.
Heath
(Adamant Media Corporation
, Boston
, 2005
). See “On floating bodies,” Book I, pp. 253
–262
; Book II, pp. 263
–300
.2.
P.
Bouguer
, Traité du Navire, de sa Construction, et de ses Mouvemens
(Jombert
, Paris
, 1746
);a summary of the Bouguer’s Treatise is in
H.
Nowacki
and L. D.
Ferreiro
, “Historical roots of the theory of hydrostatic stability of ships
,” Preprint 237, Max Planck Institute for the History of Science, Berlin, 2003
.3.
L.
Euler
, Scientia Navalis seu Tractatus de Construendis ac Dirigendis Navibus
(Petropoli Typis Academiae Scientarum
, St. Petersburg
, 1749
), Vol. 1
, Chap. 3, pp. 86
–168
, Vol. 2, Chap. 3, pp. 96–148.Available in “The works of Leonhard Euler online,” The Euler Archive, E110, E111, ⟨math.dartmouth.edu/~euler/index.html⟩.
4.
Ch.
Dupin
, Applications de Géométrie et de Mécanique, à la Marine, aux Ponts et Chaussées, etc.… Pour Faire Suite aux Développements de Géométrie
(Bachelier
, Paris
, 1822
).5.
6.
W. H.
Besant
and A. S.
Ramsey
, A Treatise on Hydromechanics. Part I: Hydrostatics
, 6th ed. (Bell
, London
, 1904
).7.
S. N.
Blagoveshchensky
, Theory of Ship Motions
, translated from the first Russian edition by Th. and L. Strelkoff, edited by L.
Landweber
(Dover
, New York
, 1962
); Original book editor: (Leningrad Shipbuilding Institute, Leningrad, 1954).8.
L. G.
Taylor
, The Principles of Ship Stability
, 5th ed. (Brown and Ferguson
, Glasgow
, 1977
).9.
E. N.
Gilbert
, “How things float
,” Am. Math. Monthly
98
, 201
–216
(1991
).10.
P.
Erdös
, G.
Schibler
, and R. C.
Herndon
, “Floating equilibrium of symmetrical objects and the breaking of symmetry, Part 1: Prisms
,” Am. J. Phys.
60
(4
), 335
–345
(1992
);P.
Erdös
, G.
Schibler
, and R. C.
Herndon
, “Part 2: The cube, the octahedron and the tetrahedron
,” Am. J. Phys.
60
(4
), 345
–356
(1992
).11.
R.
Comolet
, Mécanique Expérimentale des Fluides, T. I: Statique et Dynamique des Fluides non Visqueux
(Masson
, Paris
, 1990
).12.
13.
V.
Giles
, J. B.
Evett
, and Ch.
Liu
, Theory and Problems of Fluid Mechanics and Hydraulics
(McGraw-Hill
, New York
, 1994
).14.
15.
V.
Bertram
, Practical Ship Hydrodynamics
(Elsevier, Butterworth-Heinemann
, Oxford
, 2000
).16.
K. J.
Rawson
and E. C.
Tupper
, Basic Ship Theory
, 5th ed. (Longman
, London
, 2001
), Vol. 1
, Chap. 1–9, and Vol. 2, Chap. 10–16.17.
A.
Biran
, Ship Hydrostatics and Stability
(Elsevier, Butterworth-Heinemann
, Oxford
, 2003
).18.
D. R.
Derrett
and C. B.
Barras
, Ship Stability for Masters and Mates
, 6th ed. (Elsevier Butterworth-Heinemann
, Oxford
, 2006
).19.
J. L.
Herder
and A. L.
Schwab
, “On dynamically equivalent force systems and their application to the balancing of a broom or the stability of a shoe box
,” in Proceedings of DETC’04 ASME Design Engineering Technical Conferences and Computers and Information in Engineering Conference
[American Society of Mechanical Engineers (ASME)
, New York
, 2004
], pp. 1
–11
, ⟨audiophile.tam.cornell.edu/~als93/Publications/HerSch04.pdf⟩.20.
J.
Kliava
and J.
Mégel
, “Non-uniqueness of the point of application of the buoyancy force
,” Eur. J. Phys.
(in press).21.
Eric W.
Weisstein
, CRC Concise Encyclopedia of Mathematics
, 2nd ed. (Chapman and Hall/CRC
, Boca Raton, FL
, 2003
), p. 975
.22.
© 2010 American Association of Physics Teachers.
2010
American Association of Physics Teachers
AAPT members receive access to the American Journal of Physics and The Physics Teacher as a member benefit. To learn more about this member benefit and becoming an AAPT member, visit the Joining AAPT page.
