Floating condition – comparing buoyancy and displaced liquid weight A body will float in a fluid if the buoyant force is __________ the weight of the liquid displaced (recall Archimedes’ principle equates these two):

Introduction / Context:
Archimedes’ principle is the cornerstone of hydrostatics and naval architecture. It relates the buoyant force acting on an immersed body to the weight of the fluid displaced. This question reinforces that equality and clarifies the separate floatation condition involving the body’s own weight.

Given Data / Assumptions:

• Static fluid environment; negligible surface waves and currents.
• Body may be fully or partially submerged.
• Fluid density and gravity are uniform.

Concept / Approach:
By definition, the buoyant force F_b equals the weight of displaced fluid: F_b = weight_displaced = gamma * V_displaced, where gamma is specific weight and V_displaced is the displaced volume. Floatation occurs when F_b balances the body’s weight W_body. At equilibrium float, F_b = W_body = weight_displaced; the displaced volume adjusts to satisfy this equality.

Step-by-Step Solution:
Step 1: State Archimedes’ principle: F_b = weight of displaced fluid.Step 2: For a floating body, vertical equilibrium requires F_b = W_body.Step 3: Therefore, at float, weight of displaced liquid = W_body, and hence F_b = weight of displaced liquid (equality).Step 4: If W_body increases, the body sinks deeper to displace more fluid until equality is restored.

Verification / Alternative check:
A steel ship floats because its overall average density (including contained air) is below the water’s density. The hull submergence increases until the weight of displaced water equals the ship’s weight.

Why Other Options Are Wrong:

• Less than / more than: contradict the definitional equality between F_b and displaced fluid weight.
• Indeterminate: the relation is exact, not uncertain.
• “Opposite in direction to”: while buoyancy acts upward and weight acts downward, the question compares magnitudes to displaced liquid weight, which equals F_b.

Common Pitfalls:
Confusing “weight of displaced liquid” with “weight of the body.” The floating condition is F_b = W_body, but the magnitude of F_b is always equal to the weight of displaced fluid.

Final Answer:
equal to