Λwooden plank of length lm and uniform cross-section is hinged at one end to the bottom of the t

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Λwooden plank of length lm and uniform cross-section is...

$\Lambda$ wooden plank of length $\mathrm{lm}$ and uniform cross-section is hinged at one end to the bottom of the tank. 'Ihe tank is filled with water upto a height of $0.5 \mathrm{~m}$. 'lhe specific gravity of the plank is $0.5$. The plank makes an angle of $45^{\circ}$ with the vertical in the cquilibrium position. $A$ is the midpoint of the plank. $B$ is the middle point of the dipped parl of the plank. Which of the following are correct? (a) The weight of the plank acts downwards at $\Lambda$ (b) The buoyant force acts through $B$ (c) $O B-\frac{1}{\sqrt{2}} m$ (d) Both (b) and (c)

$\Lambda$ wooden plank of length $\mathrm{lm}$ and uniform cross-section is hinged at one end to the bottom of the tank. 'Ihe tank is filled with water upto a height of $0.5 \mathrm{~m}$. 'lhe specific gravity of the plank is $0.5$. The plank makes an angle of $45^{\circ}$ with the vertical in the cquilibrium position. $A$ is the midpoint of the plank. $B$ is the middle point of the dipped parl of the plank. Which of the following are correct?
(a) The weight of the plank acts downwards at $\Lambda$
(b) The buoyant force acts through $B$
(c) $O B-\frac{1}{\sqrt{2}} m$
(d) Both (b) and (c)

Key Concepts

Hydrostatic Pressure

Hydrostatic pressure explains how pressure in a fluid increases linearly with depth due to the weight of the overhead fluid. This concept is fundamental for understanding how forces are distributed over immersed surfaces and how pressure variations contribute to the overall buoyant force acting on a submerged object.

Center of Buoyancy

The center of buoyancy is the point through which the buoyant force, equal to the weight of the fluid displaced by an object, acts. Determining this point is crucial in problems involving submerged or partially submerged bodies, as it influences the stability and the moment balance of the system in fluid environments.

Center of Mass

The center of mass is the point at which the entire mass of an object can be considered to be concentrated. It is pivotal in analyzing the effects of gravitational forces and torques in rigid body equilibrium, as shifting the center of mass can alter the balance of forces and moments acting on the body.

Static Equilibrium

Static equilibrium refers to the condition where a body is at rest, with all forces and torques balanced. In the context of fluid mechanics and rigid body dynamics, ensuring static equilibrium involves balancing gravitational, buoyant, and reaction forces, as well as their respective moments, to prevent any net motion.

Specific Gravity

Specific gravity is a dimensionless number that represents the ratio of a material's density to that of a reference substance, typically water. It helps in determining the buoyancy of an object, as objects with a specific gravity lower than that of water tend to float, influencing the magnitude and direction of the buoyant force acting on the object.

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