Floating Bodies MCQ Quiz in বাংলা - Objective Question with Answer for Floating Bodies - বিনামূল্যে ডাউনলোড করুন [PDF]

Explanation:

Stability of Floating body:

Consider a floating body tilted by an angle Δθ, as shown below. For the untilted body the point G is the centre of gravity of the body where the body weight, W, acts. Point B is the centre of buoyancy (the centroid of the displaced volume of fluid) where the upward buoyancy force, F_B, acts. 

• When the body is tilted the centre of buoyancy moves to a new position, B', because the shape of the displaced volume changes.
• A new point, M, may be defined, called the Metacentre.
• This is the point where a vertical line drawn upwards from the new centre of buoyancy, B', of the tilted body intersects the line of symmetry of the body. The buoyancy force, F_B, now acts through B'. 
• From the centre diagram in the figure we can see that W and FB give a Restoring Moment that rotates the body back to its untitled position.
• From the right-hand diagram in the figure we can see that W and FB give an Overturning Moment that rotates the body even further in the tilted direction. 
• Hence, we can say; if the metacentre, M, lies above the centre of gravity, G, then the body is stable. In other words the Metacentric height, MG, is positive.
• If the metacentre, M, lies below the centre of gravity, G, then the body is unstable. In other words the metacentric height, MG, is negative.

​ 

∴ Option (3) is the correct answer.

Important Points

• A floating body is Stable if, when it is displaced, it returns to equilibrium.
• A floating body is Unstable if, when it is displaced, it moves to a new equilibrium. 

Explanation:

The partially submerged body resembles the floating body, where the weight of the body is balanced by the buoyancy force acting in the upwards direction. The stability of the floating is governed by the metacenter of the floating body.

Metacenter is the point about which a body starts oscillating when the body is tilted by a small angle. It is the point where the line of action of buoyancy will meet the normal axis of the body when the body is given a small angular displacement.

Stability of floating bodies:

Stable equilibrium: The metacentre is above the centre of gravity of the body, then the disturbing couple is balanced by restoring couple, the body will be in stable equilibrium.

Unstable equilibrium: The metacentre is below the centre of gravity of the body, then the disturbing couple is supported by restoring couple, the body will be in unstable equilibrium.

Neutral equilibrium: The metacenter and the centre of gravity coincides at the same point, then the body is in neutral equilibrium.

Additional Information

Stability of submerged bodies: In the case of the submerged body the centre of gravity and centre of buoyancy is fixed, therefore the stability or unstability is decided by the relative positions of the centre of buoyancy and the centre of gravity.

Stable equilibrium: For stable equilibrium, the body the centre of buoyancy is above the centre of gravity. The disturbing couple is countered by the restoring couple.

Unstable equilibrium: For unstable equilibrium, the centre of buoyancy is below the centre of gravity of the body, the disturbing couple is supported by the restoring couple.

Neutral equilibrium: when the centre of gravity and the centre of buoyancy coincide then it is the state of neutral equilibrium.

Explanation:

Metacentre:

The point ' M ' at which the line of action of the new buoyant force intersects the original vertical through the CG of the body, is called the metacentre. It is a point about which a floating body starts oscillating when given a small angular displacement.

Concept:

The partially submerged body resembles the floating body, where the weight of the body is balanced by the buoyancy force acting in the upwards direction. The stability of the floating is governed by the metacenter of the floating body.

Metacenter is the point about which a body starts oscillating when the body is tilted by a small angle. It is the point where the line of action of buoyancy will meet the normal axis of the body when the body is given a small angular displacement.

Stability of floating bodies:

Stable equilibrium: Metacenter is above the center of gravity of the body, then the disturbing couple is balanced by restoring the couple, the body will be in stable equilibrium.

Unstable equilibrium: Metacenter is below the center of gravity of the body, then the disturbing couple is supported by restoring the couple, the body will be in unstable equilibrium.

Neutral equilibrium: The metacenter and the center of gravity coincides at the same point, then the body is in neutral equilibrium.

Stability of submerged bodies

In the case of the submerged body, the center of gravity and center of buoyancy is fixed, therefore the stability or instability is decided by the relative positions of the center of buoyancy and the center of gravity.

Stable equilibrium: For stable equilibrium, the body the center of buoyancy is above the center of gravity. The disturbing couple is countered by the restoring couple.

Unstable equilibrium: For unstable equilibrium, the center of buoyancy is below the center of gravity of the body, the disturbing couple is supported by the restoring couple.

Neutral equilibrium: When the center of gravity and the center of buoyancy coincides then it is the state of neutral equilibrium.

Concept:

Principle of floatation: 

For a body floating in a fluid, the weight of the body must be balanced by buoyant force.

Weight (W) = Buoyant force (FB) = ρ × g × V

where, ρ = density of the fluid, V = Volume of immersion

Calculation:

Given:

L = 6 m, B = 2.5 m, and H = 1.5 m

Depth of immersion = y m

For a body to float, weight should be equal to buoyant force.

W = FB

FB = Density of water (ρ) × g × volume of immersion (V)

650 × g × (6 × 2.5 × 1.5) = 1000 × g × (6 × 2.5 × y)

\(y=\frac{650 × 6 × 2.5 × 1.5}{1000 × 6 × 2.5}\)

So, y = 0.975 m

Concept:

Metacentric height (GM)

⇒ +ve ⇒ stable equilibrium

⇒ -ve ⇒ unstable equilibrium

⇒ 0 ⇒ Neutral equilibrium

\(GM = \frac{I}{\forall } - BG\)

where,

G → centre of mass/gravity, B → centre of buoyancy, M → Metacentre, GM → Metacentric height

Calculation:

For a cylinder,

x = 0.6 H (Equating Bouyancy force with weight)

and \(BG = \frac{1}{2}\left[ {H - x} \right]\)

∴ we can write,

x = 0.6 H = 0.6 × 2d = 1.2 d

\(BG = \frac{1}{2}\left[ {H - x} \right]\)

\(BG = \frac{1}{2}\left[ {2d - 1.2\;d} \right] = 0.4\;d\)

\(\therefore GM = \frac{{\frac{\pi }{{64}}{d^4}}}{{\frac{\pi }{4}{d^2}x}} - 0.4\;d\)

GM = 0.052 d – 0.4 d

GM = - 0.3479 ⇒ Negative

∴ Unstable equilibrium

Concept:

• Archimedes Principle: The upward buoyant force that is exerted on a body immersed in a fluid, whether partially or fully submerged, is equal to the weight of the fluid that the body displaces and acts in the upward direction at the centre of mass of the displaced fluid.
  • The value of thrust force is given by the Archimedes law which was discovered by Archimedes of Syracuse of Greece. When an object is partially or fully immersed in a liquid displaced by it.
  • Archimedes' principle tells us that this loss of weight is equal to the weight of liquid the object displaces. If the object has a volume of V, then it displaces a volume V of the liquid when it is fully submerged. If only a part of the volume is submerged, the object can only displace that much liquid.
• Archimedes Principle Formula: In simple form, the Archimedes law states that the buoyant force on an object is equal to the weight of the fluid displaced by the object. Mathematically written as:

Fb =  ρ × g × V

Where Fb is the buoyant force, ρ is the density of the fluid, V is the submerged volume, and g is the acceleration due to gravity

• When a body partially and fully emerged in any fluid it experiences an upward force that is equal to the weight of the fluid displaced by it and it is called buoyant force.
• Under equilibrium conditions, the Buoyant force is equal to the weight of the body.
• Weight of liquid displaced = Weight of the body

Explanation:

Density of sea water (ρsw) > density of river water  (ρrw)

For the same ship, balancing the buoyant force for both conditions,

\({\rho _{sw}} \times g \times {h_{sw}} = {\rho _{rw}} \times g \times {h_{rw}}\)

\({\rho _{rw}} < {\rho _{sw}} \Rightarrow {h_{sw}} < {h_{rw}},\;\) where h is the depth of ship into the water

∴ The ship rises up when it enters the sea from a river.

Explanation:

When a body is submerged fully or partially in any fluid it will experience an upward force, called buoyant force, which is equal to the weight of the fluid displaced by that body.

Iceberg is floating in the water, this means that the upward buoyant force on a portion of an iceberg submerged in seawater is balanced by the downward weight of the iceberg.

Explanation:

• Meta Centre is defined as the point about which a body starts oscillating when body is tilted by a small angle.
• The meta-centre may also be defined as the point at which the line of action of the force of buoyancy will meet the normal axis of the body when the body is given a small angular displacement.
• The distance MG i.e. the distance between the meta-centre of a floating body and the centre of gravity of the body is called meta-centric height. 
• It is measured along the line BG.