**Buoyancy and Floatation**

**Buoyancy**

When a body is immersed in a fluid, an upward force is exerted by the fluid on the body which is equal to the weight of the fluid displaced by the body. This upward force is called buoyant force and the phenomenon is called buoyancy.

**Archimedes Principle**

- When a body is submerged either
*fully*or*partially*then it is acted upon by a*Force of Buoyancy*in the vertical direction which is equal to the weight of the liquid displaced by the body. - This Buoyant Force always acts through the centroid of liquid displaced.
- The center of Buoyancy is the point through which buoyant force acts.

**Principle of Flotation**

- According to this principle, if the weight body is equal to the buoyant force then, the body will float.

**Hρ _{Body} = hρ_{fluid}**

H = Height of Body

h = height of body submerged in fluid

**Centre of Buoyancy**

- The point at which the force of buoyancy acts is called the center of buoyancy.
- It lies at the center of gravity (G) of the volume of water displaced.

**Metacenter**

If a body that is floating in a liquid is given small angular displacement, it starts oscillating about some point, ‘M’. This point is called the Metacenter.

**Metacentric Height (GM or MG)**

It is the distance between the gravity center and the metacenter.

**Check Also: Fluid Mechanics: Objective Questions**

**Equilibrium ****Condition ****for Floating and Submerged Body**

**For Stable Equilibrium**

- In the case of a floating body, the
*metacenter*should be*above*the*center of gravity*. - In the case of a submerged body, the
*center of buoyancy (B)*should be*above*the*center of gravity (G).*

**For Unstable Equilibrium**

- In the case of a floating body, the
*Metacenter (M)*lies*below*the*Center of Gravity (G).* - In the case of a submerged body, the
*Center of Buoyancy (B)*lies*below the Center of Gravity (G).*

**For Neutral Equilibrium**

- In the case of a floating body, M and G both coincide.
- In the case of a submerged body, ‘B’ and ‘G’ both coincide.

**Distance between Metacenter and Center of Buoyancy**

BM = I_{min}/V_{immersed}

Where I_{min} = Moment of Inertia of the top view of the floating body about the longitudinal axis

V = Volume of the body immersed in liquid

**Relation between B, G, and M**

**The relationship between the Center of Buoyancy (B), the Center of Gravity (G), and the Metacenter (M) is given by**

GM = I/V – BG

BG = Distance between CG of the whole body and CG of the submerged part.

**Also, If**

- GM > 0 (Stable Equilibrium)
- GM < 0 (Unstable Equilibrium)
- GM = 0 (Neutral Equilibrium)

**Remember-**

- The metacenter height for the rolling condition will be less than the metacentric height for the pitching condition.

**Time Period of Oscillation**

- If a floating body oscillates then its time period of transverse oscillation is given by

T = 2\pi \sqrt { \frac { { K }_{ G }^{ 2 } }{ g\times GM } }

GM = Metacentric height

{ K }_{ G }^{ 2 } = Least Radius of Gyration