Page 32 - Mechanical Engineers Reference Book

Mechanics of fluids 6/21

Projected
area A,

IGP
I
I CP
-+ ~
Liquid
surface

(a) Concave curved surface
Free surface k< __ j

Figure 1.37 Stability
(b) Convex surface
bodies in (d) and (g). A body is in unstable equilibrium if a
Figure 136 small displacement produces a disturbing force or moment as
for the ball in Figure 1.37(b) or the floating bodies (e) and (h).
A body is in neutral equilibrium if a small displacement
the body. The first recognition of this is attributed to Archi- produces no force or moment as for the ball in Figure 1.37(c)
medes.
or the floating bodies in (f) and (i).
For a partially immersed body, the point at which the line of
(a) Displacement force The buoyancy or displacement force action of the buoyancy force FB cuts the vertical centre line of
FB on a body fully or partially immersed in a fluid is equal to the floating body in the displaced positior, is known as the
the weight of the volume of the fluid equivalent to the metacentre (M). For a floating body to be stable M must lie
immersed volume of the body (the weight of the displaced above the body’s centre of gravity, G. If M lies below G the
volume 17, of the fluid):
body is unstable; if M lies on G the body is in neutral
FB = %I’D (1.19) equilibrium. The distance GM is known as the metacentric
height. The distance of the metacentre above the centre of
This buoyancy force acts vertically upwards through the buoyancy
centroid of the displaced volume, which is known as the centre
of buoyancy (19). If the buoyancy force is equal to the weight I
of the body then the body will float in the fluid. If the weight BM=- (1.20)
of the basdy is greater than the buoyancy force then the body VD
will sink. If the buoyancy force is greater than the weight of
the body then the body will rise. where I = second moment of area of the body at the water
In a liquid, for example, a body will sink until the volume of line (liquid surface) about its central axis normal to the
liquid dkplaced has a weight which is equal to that of the direction of displacement.
body. If the body is more dense than the liquid then the body
will not float at any depth in the liquid. A balloon will rise in (c) Period of oscillation of a stable floating body A floating
air until the density of the air is such that the weight of the body oscillates with the periodic time T of a simple pendulum
displaced volume of air is equal to the weight of the balloon. of length k21GM, where k is the radius of gyration of the body
about its axis of rotation. The periodic time is given by
(b) Stability of a Poating body Figure 1.37 shows bodies in
various stages of equilibrium. A body is in stable equilibrium if 0.5
a small displacement produces a restoring force or moment as T = 24&) (1.21)
for the ball in the saucer in Figure 1.37(a) or the floating