Bodies for equilibrium r and w must be equal and act

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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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