Fundamental concepts of the stability and instability explained using “ball on the floor” analogy.
Stable since any small disturbance generates a restoring force (due to gravity) that returns it to its initial position.
Neutrally stable because if any small disturbance, it would stay put at its new location. It has no tendency to move back to its original location, nor does it continue to move away.
Unstable is a situation in which the ball may be at rest at the moment, but any disturbance, even an infinitesimal one, causes the ball to roll off the hill—it does not return to its original position; rather it diverges from it
It is not appropriate to discuss stability if the object is not in a state of equilibrium. For Immersed or floating body in static in static equilibrium, the weight and the buoyant force acting on the body balance each other, these bodies inherently stable vertically.
A floating body possesses vertical stability, while an immersed neutrally buoyant body is neutrally stable since it does not return to its original position after a disturbance.
The Rotational Stability Criteria of an Immersed Body
G-center of gravity
B-center of buoyancy
W-weight
FB-buoyant force
Figure (a) shows the body is stable due to the heavy bottom, thus point G is directly below point B. A rotational disturbance of the body in such cases produces a restoring moment to return the body to its original stable position. Therefore, a stable design for a submarine calls for the engines and the cabins for the crew to be located at the lower half in order to shift the weight to the bottom as much as possible.
Figure (b) shows a body for which G and B coincide is neutrally stable. This is the case for bodies whose density is constant throughout. For such bodies, there is no tendency to overturn or right themselves.
Figure (c) shows an immersed body G is directly above point B is unstable, and any disturbance will cause this body to turn upside down.
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The Rotational Stability Criteria for Floating Bodies
The floating body is bottom-heavy and thus the center of gravity G is directly below the center of buoyancy B, the body is always stable. But unlike immersed bodies, a floating body may still be stable when G is directly above B as shown before.
This is because the centroid of the displaced volume shifts to the side to a point B’ during a rotational disturbance while the center of gravity G of the body remains unchanged. If point B is sufficiently far, these two forces create a restoring moment and return the body to the original position
A measure of stability for floating bodies is the metacentric height GM, which is the distance between the center of gravity G and the metacenter M—the intersection point of the lines of action of the buoyant force through the body before and after rotation.
A floating body is stable if point M is above point G, and thus GM is positive, and unstable if point M is below point G, and thus GM is negative. In the latter case, the weight and the buoyant force acting on the tilted body generate an overturning moment instead of a restoring moment, causing the body to capsize. The length of the metacentric height GM above G is a measure of the stability: the larger it is, the more stable is the floating body.
A floating body is stable if the body is bottom-heavy and thus the center of gravity G is below the centroid B of the body, or if the metacenter, M is above point G. However, the body is unstable if point M is below point G.
FLUIDS MECHANICS 