Step-by-Step Explanation
- Buoyant Force Equation: Archimedes’ principle states that the buoyant force (F_b) is equal to the weight of the displaced fluid. This can be written as:
F_b = ρ_fluid × V_displaced × g - Weight of the Object: The weight of the object (W) is given by:
W = ρ_object × V_object × g - Equilibrium Condition for Floating: For an object to float, the buoyant force must equal the weight of the object:
F_b = W - Substitute the Expressions: Substitute the expressions for F_b and W from steps 1 and 2 into the equilibrium condition:
ρ_fluid × V_displaced × g = ρ_object × V_object × g - Cancel Out Gravity (g): Since the gravitational acceleration (g) is the same on both sides of the equation, we can cancel it out:
ρ_fluid × V_displaced = ρ_object × V_object - Solve for the Submerged Volume: Recognize that for a floating object, the displaced volume of the fluid (V_displaced) is equal to the submerged volume of the object (V_submerged). Thus, we have:
ρ_fluid × V_submerged = ρ_object × V_object - Find the Submerged Fraction: To find the fraction of the object that is submerged, divide both sides by the total volume of the object (V_object):
V_submerged / V_object = ρ_object / ρ_fluid
Conclusion By canceling out the gravitational term (g) from both sides of the equation, we see that the fraction of the object’s volume that is submerged depends only on the ratio of the object’s density to the fluid’s density. This simplification shows that the condition for floating is independent of the gravitational acceleration, as long as the fluid and the object are subject to the same gravitational field.