Question

A solid body of uniform specific gravity floats in a deep liquid pool. Take $B, G$, and $M$ as the centre of buoyancy, centre of gravity, and metacentre of the body, respectively. Which one of the following options is correct for the stable floatation of the body in the pool when the body is given a small tilt angle?

• A. $MG$ is the metacentric height and $G$ should lie below $M$
• B. $MG$ is the metacentric height and $B$ should lie above $M$
• C. $MB$ is the metacentric height and $B$ should lie below $M$
• D. $MB$ is the metacentric height and $G$ should lie above $M$

Hint:

For floating bodies, always check the relative positions of $M$ and $G$. Stability requires $M$ above $G$, and metacentric height is $MG$.

Answer 1

The Correct Option is A

Step 1: Definitions.
- $B$: Centre of buoyancy (geometric centroid of displaced liquid volume). - $G$: Centre of gravity of the floating body. - $M$: Metacentre (intersection point of buoyant force line when body is tilted with vertical through $B$ in equilibrium).

Step 2: Concept of metacentric height.
Metacentric height is defined as: \[ GM = BM - BG, \] where $BM$ is distance between buoyancy centre and metacentre, $BG$ is distance between buoyancy centre and centre of gravity. Thus, metacentric height is the distance between $G$ and $M$: \[ MG = \text{Metacentric height}. \]

Step 3: Condition for stability.
For stable equilibrium of floating bodies: - The metacentre $M$ must lie above the centre of gravity $G$. - This ensures that when the body tilts slightly, the buoyant force provides a restoring couple. If $G$ is above $M$, then the couple is overturning $\Rightarrow$ unstable equilibrium.

Step 4: Eliminate wrong options.
- (B) says $B$ should lie above $M$ — incorrect, because stability condition involves $G$ relative to $M$. - (C) and (D) call $MB$ the metacentric height — incorrect, the true definition is $MG$. Final Answer:
\[ \boxed{MG \text{ is the metacentric height and } G \text{ should lie below } M} \]