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Unformatted text preview: ibrium of Floating Bodies
• For equilibrium,
• R and W must be equal and
act in the same straight line
Body weight, W = mg
Where,
See EXAMPLE 1
G
W = mg
B
R = weight of fluid displaced
Volume
= ρgV
Upthrust, R displaced, V
Therefore,
ρgV = mg
V = m/ρ Stability of A
Submerged Body Stability of A Submerged Body
• For body totally
immersed in a fluid,
• W acts through the
centre of gravity of the
body
• R acts through the
centroid of the body
(centre of buoyancy) R
B
G W
W
G
B R Stability of A Submerged Body
θ
R R • If G is below B,
• A small displacement θ
will generate a righting
moment, and
• The body will return to
its equilibrium position B B G W
W
G
B R G W Righting
moment
θ R
B G W Stability of A Submerged Body
θ
R R • If G is above B,
• A small displacement θ
will produce an
overturning moment,
and
• The body is unstable B B G G W W
W
G
B R θ W
G
B R Overturning
moment Stability of
Floating Bodies Stability of Floating Bodies
W = mg • A body floating in equilibrium
• W acts through the
centre of gravity
• R acts through the
centre of buoyancy B of the
displaced fluid in the same
straight line as W G
B R=W W = mg
G
B R=W Stability of Floating Bodies
• When body is displaced
through an angle θ,
• W continue to act
through G, and
• The volume of liquid
remains unchanged
since R = W
• But the shape of the
volume changes, and
• Its centre of buoyancy
moves relative to the
body from B to B1 W = mg
G W = mg
G
B B R=W R=W
θ θ M
G
M B1 x
RW G
B1 x
WR Stability of Floating Bodies
• Since R and W are no
longer in the same
straight line,
• A turning moment is
produced = W x GM x θ
x = GM x θ W = mg W = mg
G G B B R=W Overturning θ
(for small θ, sin θ = θ in radians)
moment
G
M B1 x RW R=W
θ
Righting
moment M G
B1 x
WR Stability of Floating Bodies
W = mg W = mg
G G B B • M is the point at which
R=W
the line of action of
the upthrust R cuts
Overturning θ
the original vertical
moment
through the centre of
G
gravity G
M
x
B1 RW R=W
θ
Righting
moment M G
B1 x
WR Stability of Floating Bodies W = mg • The point M is called
Metacentre, and
G
B
• Distance GM is the
metacentric height
R=W
• If M lies above G, a
above
righting moment
righting
θ
W x GM x θ is produced,
• Equilibrium is stable,
stable
M
and
G
• GM is regarded as
B1
positive
positive
Righting
moment θ
M
G
B1 x
WR Stability of Floating Bodies
• If M lies below G, an
If
below G,
overturning moment
overturning
W x GM x θ is pr...
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 Fall '11
 LamersAKBP

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