Static Stability of Floating Body
1.0 ) THEORY
(I) Static Stability of Floating Body
Referring to Figure 1, the weight W of the floating body passes through its center of gravity G. The upthrust or buoyancy force FB acting on the floating body passes through the center of buoyancy B, which corresponds to the centroid of the displaced fluid. When the floating body is subjected to a small angular displacement or perturbation θ about its equilibrium upright configuration, the center of buoyancy shifts from B to B’, while the center of gravity of the floating body remains unchanged at G. A vertical line drawn upward from B’ intersects the line of symmetry at M, known as the metacenter. GM is known as the metacentric height.
(a) If M is above G (GM > 0), a restoring couple acts on the floating body in its displaced position tending to restore it to its original position. Hence, the body is in stable equilibrium.
(b) If M is below G (GM < 0), an overturning couple acts on the body. Hence, the body is in unstable equilibrium.
(c) If M coincides with G (GM = 0), the resultant couple is zero, and the body has no tendency to return to, nor move further away from its original position. Hence, the body is in neutral equilibrium.
Figure 1 Static stability of a floating body
If the body floats stably, it may be shown that the period of oscillations for small angles of displacement θ is given by , where KOy is the radius of gyration of the floating body about its longitudinal axis.
Hence, larger values of GM give rise to more rapid oscillations, thus subjecting passengers to higher levels of discomfort. However, the larger the value of GM, the more stable the floating body is. The above are two conflicting requirements for the choice of GM. A good design should thus entail adequate but not excessive values of GM.
(II) Inclined Experiment to Determine GM and KG
In this experiment, a mass m is moved transversely across the deck through a known distance d, as shown in Figure 2.
(I) Static Stability of Floating Body
Referring to Figure 1, the weight W of the floating body passes through its center of gravity G. The upthrust or buoyancy force FB acting on the floating body passes through the center of buoyancy B, which corresponds to the centroid of the displaced fluid. When the floating body is subjected to a small angular displacement or perturbation θ about its equilibrium upright configuration, the center of buoyancy shifts from B to B’, while the center of gravity of the floating body remains unchanged at G. A vertical line drawn upward from B’ intersects the line of symmetry at M, known as the metacenter. GM is known as the metacentric height.
(a) If M is above G (GM > 0), a restoring couple acts on the floating body in its displaced position tending to restore it to its original position. Hence, the body is in stable equilibrium.
(b) If M is below G (GM < 0), an overturning couple acts on the body. Hence, the body is in unstable equilibrium.
(c) If M coincides with G (GM = 0), the resultant couple is zero, and the body has no tendency to return to, nor move further away from its original position. Hence, the body is in neutral equilibrium.
Figure 1 Static stability of a floating body
If the body floats stably, it may be shown that the period of oscillations for small angles of displacement θ is given by , where KOy is the radius of gyration of the floating body about its longitudinal axis.
Hence, larger values of GM give rise to more rapid oscillations, thus subjecting passengers to higher levels of discomfort. However, the larger the value of GM, the more stable the floating body is. The above are two conflicting requirements for the choice of GM. A good design should thus entail adequate but not excessive values of GM.
(II) Inclined Experiment to Determine GM and KG
In this experiment, a mass m is moved transversely across the deck through a known distance d, as shown in Figure 2.
References: 1. Jump up to:a b "Archimedes (Greek mathematician) - Britannica Online Encyclopedia". Britannica.com. Retrieved 2012-08 2. Jump up to:a b "The works of Archimedes". p. 257. Retrieved 11 March 2010. "Any solid lighter than a fluid will, if placed in the fluid, be so far immersed that the weight of the solid will be equal to the weight of the fluid displaced." 3. "On Floating Bodies (Book II)". Math.nyu.edu. Retrieved 2012-08-13. 4. Zhukovskii, N. E. Teoreticheskaia mekhanika, 2nd ed. Moscow-Leningrad, 1952.