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When a body, floating in a liquid is given a small angular displacement, it starts oscillating about a point known as
Where M is metacentre and θ is small angle of hill through which a body is tilted.
Hence metacenter is a point about which a body oscillate when tilted or given a small angular displacement.
The time of oscillation of a floating body is given by
Where
K = Radius of Gyration of the floating body about its centre of gravity
h = Metacentric height of the floating body
Time of oscillation of a floating body is
K = radius of gyration
h = metracentric height
The time of oscillation of a floating body with increase in metacentric height will be
Time of oscillation
as h increases, time of oscillation decreases
For stable equilibrium
A submerged body will be in stable equillibrium. if
For stable equilibrium of submerged body:
Hence for stable equilibrium of submerged body centre of buoyancy B should be above G.
Which one of the following is the conditions for stable equilibrium for a floating body
For stable equilibrium
For stable equilibrium metacenter is above the center of gravity.
How is the metacentric height (GM) expressed
Where I = Moment of inertia of the plan of the floating body at the water surface
V = Volume of the body submerged in water
BG = Distance between the center of gravity (G) and the center of Buoyancy (B)
Metacentric height,
GM = BM  BG
Where I = Area moment of inertia of top view about longitudinal axis
V = Volume of the fluid displaced
Hence,
An odd shaped body weighing 7.5 kg and occupying 0.01 m^{3} volume wiil be completely submerged in a fluid having specific gravity of
A cylindrical body of crosssectional area A height H and density ρ_{s} is immersed to a depth h in a liquid of density ρ and tied down to bottom with a string then the tension in the string is
w + T = F_{B} = weight of fluid displaced
A float valve of the ‘ballcock’ type required to close an opening of a supply pipe feeding a cistern as shown in the given figure
The buoyant force F_{B} required to be exerted by the float to keep the valve closed against a pressure of 0.28 N/mm^{2} is
Taking moment about hinge We get
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