Question

When a body is floating in the liquid and given a small angular displacement, then the body starts oscillating about a centre. That centre is called as

• A. Centre of pressure
• B. Centre of gravity
• C. Centre of buoyancy
• D. Metacentre

Hint:

Metacentric Height (\(GM\)) analytical formula: \[ GM = \frac{I}{V_{\text{submerged}}} - BG \] Where: - \(I\) = Second moment of area of the floating body at the water surface plane. - \(V_{\text{submerged}}\) = Total volume of fluid displaced. - \(BG\) = Distance between the center of buoyancy and center of gravity.

Answer 1

The Correct Option is D

Concept: The stability of a floating body depends on the relative positions of its geometric and gravitational centers, as well as the behavior of the buoyant force when the body tilts. Let us define the primary reference centers for a floating body at equilibrium:
• Centre of Gravity (\(G\)): The single point through which the total gravitational weight force (\(W\)) acts vertically downward.
• Centre of Buoyancy (\(B\)): The centroid of the displaced volume of liquid, through which the buoyant force (\(F_B\)) acts vertically upward. At static equilibrium, \(G\) and \(B\) lie on the same vertical axis of symmetry.

Step 1: Analyzing the shift in centers during angular displacement.
When the floating body is given a small angular tilt or heel angle \(\theta\), its underwater geometry becomes asymmetric. The submerged volume shifts toward the tilting side, moving the centroid of this displaced liquid volume to a new position, denoted as \(B'\).

Step 2: Defining the Metacentre (\(M\)).
Because the center of buoyancy shifts to \(B'\), the line of action of the buoyant force shifts as well. The vertical line acting upward through the new center of buoyancy \(B'\) intersects the original vertical axis of symmetry of the body at a specific point. This point of intersection for small angles of tilt (\(\theta \to 0\)) is defined as the Metacentre (\(M\)).

Step 3: Characterizing the oscillatory behavior.
As the body rolls or tilts slightly, the restoring or upsetting couple acts about a pivot point. The metacentre \(M\) serves as this effective rotational center about which the floating body begins to oscillate.
• If \(M\) lies above \(G\) (positive metacentric height, \(GM > 0\)), a restoring torque returns the body to equilibrium, resulting in stable oscillations about \(M\).
• If \(M\) lies below \(G\) (\(GM < 0\)), the body is unstable and will capsize. Thus, the center about which the body oscillates is the metacentre, which corresponds to Option (4).