Wood of density 0.8: submerged fraction

A wooden block with a density of 0.8 grams per cubic centimeter floats in water with a density of 1.0 grams per cubic centimeter. Find the ratio of the submerged volume to the total volume of the block.

Step-by-step

1. V_sub/V = ρ_object/ρ_fluid = 0.8/1 = 80%
   When floating, buoyancy equals weight: ρ_fluid·g·V_sub = ρ_object·g·V
2. ρ_object = 0.8 , ρ_fluid = 1
   Comparing the object's density with the liquid's tells you whether it floats

Why it works

The condition for floating is that buoyant force equals gravitational force. Expanding both sides shows that the ratio of liquid density to object density equals the ratio of submerged volume to total volume. This ratio depends only on the two densities, not on the size of the object, so objects made of the same material always have the same immersion fraction regardless of their size. A density ratio less than one indicates floating, equal to one indicates neutral buoyancy (suspended), and greater than one indicates sinking.

Common mistakes

• Writing the immersion ratio as liquid density divided by object density, taking the reciprocal.
• Mixing density units and directly dividing grams per cubic centimeter with kilograms per cubic meter.
• Confusing "submerged volume fraction" with "exposed volume fraction".

Related

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