Archimedes’ law of buoyancy – magnitude of the buoyant force For a body immersed wholly or partially in a fluid at rest, the buoyant force acting on the body is always __________ the weight of the fluid displaced by the body.

Introduction:
Buoyancy is a cornerstone concept in fluid mechanics and naval architecture. The question tests the precise statement of Archimedes’ law regarding the magnitude of the buoyant force on a body immersed in a static fluid. Understanding this equality is essential for predicting floatation, apparent weight loss, and stability analyses.

Given Data / Assumptions:

• Fluid at rest (hydrostatics).
• Body may be fully or partially submerged.
• Gravitational field is uniform; fluid density is constant.
• No fluid motion–induced dynamic effects (no lift or drag).

Concept / Approach:

Archimedes’ principle states: the buoyant force on a submerged body equals the weight of the fluid displaced by that body. This arises from integrating the hydrostatic pressure over the body's surface. Pressure increases with depth, producing a net upward resultant equal to rho * g * V_displaced, whose weight is rho * g * V_displaced. Hence, the magnitude of buoyant force equals the weight of fluid displaced, independent of the body's own material (though floatation depends on comparing forces).

Step-by-Step Solution:

Verification / Alternative check:

For a floating object in equilibrium, weight of body W equals buoyant force, so W = rho * g * V_displaced. For a fully submerged, tethered body, the scale reading equals W − F_buoyancy, again showing F_buoyancy equals the displaced fluid weight irrespective of the body's density in still fluid.

Why Other Options Are Wrong:

Less than / more than: Contradict the exact equality from Archimedes’ law.Dependent on body shape only: Shape affects displaced volume but not the equality relation.Zero unless floating: Even a fully submerged object experiences buoyancy.

Common Pitfalls:

Confusing buoyant force with net force; net force depends on comparing buoyancy with the object’s weight. Also, mixing hydrostatic buoyancy with dynamic lift that occurs in moving fluids.

Final Answer:

equal to