Floating and sinking: compare the two densities
Definition
Density
Mass per unit volume, ρ=m/V\rho = m/Vρ=m/V, with unit kg/m3\text{kg/m}^3kg/m3 or g/cm3\text{g/cm}^3g/cm3.
- Everything here comes from one comparison: the density of the object against the density of the fluid it is placed in.
- If the object is less dense than the fluid, it floats.
- If the object is denser than the fluid, it sinks.
- If the two densities are equal, the object stays where it is put, neither rising nor sinking.
- A floating object settles into the fluid only until it has pushed aside a weight of fluid equal to its own weight: a less dense object can do this with part still above the surface, while a denser object cannot even when fully submerged, so it keeps going down.
- Worth memorising: water is 1000 kg/m3=1.0 g/cm31000\ \text{kg/m}^3 = 1.0\ \text{g/cm}^31000 kg/m3=1.0 g/cm3, and air is about 1.2 kg/m31.2\ \text{kg/m}^31.2 kg/m3.
Matching the units before you compare
- The comparison is only valid if both densities are in the same unit, so convert first and compare second.
- Using 1 g/cm3=1000 kg/m31\ \text{g/cm}^3 = 1000\ \text{kg/m}^31 g/cm3=1000 kg/m3, a density of 800 kg/m3800\ \text{kg/m}^3800 kg/m3 is 0.8 g/cm30.8\ \text{g/cm}^30.8 g/cm3, which is less than water's 1.0 g/cm31.0\ \text{g/cm}^31.0 g/cm3, so that object floats.
Objects that are not solid all the way through
- When an object contains air spaces, such as a steel ship, a sealed bottle or a lifejacket, use the average density of the whole object: total mass divided by total outside volume.
- A steel ship floats even though steel is about eight times denser than water, because most of the volume enclosed by its hull is air.
Which liquid floats on which
- Two liquids that do not mix (immiscible, like oil and water) settle into separate layers.
- The less dense liquid floats on top of the denser one, and with three or more liquids the layers stack in order of increasing density downwards.
- Density data is usually given as masses and volumes, so calculate each density with ρ=m/V\rho = m/Vρ=m/V first, then rank them.
Common Mistake
- Floating is decided by density, not by mass or weight. "It is heavy so it sinks" earns nothing: a steel ship is far heavier than a steel nail, yet it is the ship that floats.
- Do not compare a density in kg/m3\text{kg/m}^3kg/m3 with one in g/cm3\text{g/cm}^3g/cm3. Convert both to the same unit first; forgetting the factor of 100010001000 is the commonest error here.
- A floating object still has weight. It floats because the resultant force on it is zero, not because gravity has stopped acting.
- For a hollow object, use the total mass over the total volume, not the density of the material it is made from.
- The liquid-layer result only works because the liquids do not mix; say so if asked to justify your answer.
Exam technique
- Write the comparison down, not just the conclusion: "density of the block =0.80 g/cm3= 0.80\ \text{g/cm}^3=0.80 g/cm3, less than water's 1.0 g/cm31.0\ \text{g/cm}^31.0 g/cm3, so it floats."
- Quote ρ=m/V\rho = m/Vρ=m/V from recall, substitute with consistent units, and attach the unit to the density.
- If a multiple-choice item gives four densities, convert them all to one unit before ranking them.
Self review
- What decides whether an object floats or sinks in a fluid?
- What happens to an object whose density equals the density of the fluid?
- Why does a steel ship float when a solid lump of steel sinks?
- Water is 1.0 g/cm31.0\ \text{g/cm}^31.0 g/cm3; will an object of density 850 kg/m3850\ \text{kg/m}^3850 kg/m3 float on it?
- When two liquids that do not mix are poured together, which one forms the top layer?
