Density, specific gravity: Question bank — Physics deep dive

Recall the two tools we lean on throughout:

Density $ρ = m / V$ — mass packed per unit of space (see Density, specific gravity).
Specific gravity $SG = ρ_{substance} / ρ_{water}$ , a pure number with $ρ_{water} = 1000 kg/m^{3}$ .

True or false — justify

Recall Verdict

and reason required A heavier object is always denser than a lighter one. ::: False. Density is mass per volume; a huge light log outweighs a pebble yet floats because its mass is spread over far more space. Doubling the size of an iron cube doubles its density. ::: False. Both mass and volume grow by $8 \times$ (side cubed), so their ratio $ρ = m / V$ is unchanged — density is intrinsic to the material. Specific gravity has the same units as density. ::: False. It is a ratio $\frac{kg/m ^{3}}{kg/m ^{3}}$ , so the units cancel and SG is a pure dimensionless number. If SG $= 1$ exactly, the object floats fully submerged with its top just level with the surface. ::: True. Equal densities mean weight and maximum buoyancy balance at 100% submersion, so it hangs neutrally at any depth — neither rising nor sinking. See Buoyancy and Archimedes' principle. A cup of seawater is denser than a cup of pure water. ::: True. Dissolved salt adds mass to nearly the same volume, so $ρ$ rises (about $1025 kg/m^{3}$ ) — which is why floating is easier in the sea. Cutting a block in half halves its specific gravity. ::: False. SG depends on the material, not the amount; each half has the same $ρ$ and therefore the same SG. Water at $4^{\circ} C$ is used as the reference because it is the coldest water can get. ::: False. Water can be colder (down to $0^{\circ} C$ before freezing); $4^{\circ} C$ is chosen because water is at its densest there, giving a clean, reproducible standard. The SG of gold ( $19.3$ ) equals its density in $g/cm^{3}$ . ::: True. Since water is $1 g/cm^{3}$ , dividing any density in $g/cm^{3}$ by $1$ leaves the number unchanged — that numerical coincidence is why the two match.

Spot the error

Recall Find the flawed step, then state the fix

"Mixing equal volumes of $ρ_{1}$ and $ρ_{2}$ , I average the densities: $\frac{ρ _{1} + ρ _{2}}{2}$ ." ::: Correct here — for equal volumes the arithmetic mean is right, since equal $V$ makes each density contribute equally to total mass. The trap is copying this for equal masses. "Mixing equal masses of $ρ_{1}$ and $ρ_{2}$ , I average the densities." ::: Error. For equal masses the denser liquid takes up less volume, so you must use the harmonic mean $\frac{2 ρ _{1} ρ _{2}}{ρ _{1} + ρ _{2}}$ ; a plain average over-counts the denser one. "Oil has SG $= 0.8$ , so its density is $0.8 kg/m^{3}$ ." ::: Error. SG is a ratio; multiply by water: $ρ = 0.8 \times 1000 = 800 kg/m^{3}$ . Forgetting the $\times 1000$ is the classic slip. " $1 g/cm^{3} = 1 kg/m^{3}$ because I just swapped the units." ::: Error. Grams to kg is $\times 1 0^{- 3}$ but $cm^{3}$ to $m^{3}$ is $\times 1 0^{- 6}$ ; the net factor is $1 0^{3}$ , so it is $1000 kg/m^{3}$ . "Ice sinks in water because ice is a solid and solids are denser than liquids." ::: Error. Solids are usually denser than their own liquid, but water is a rare exception — ice ( $917 kg/m^{3}$ ) is less dense than liquid water, so it floats. "Since pressure is $P = ρ g h$ , deep water is denser than shallow water." ::: Error. In the $P = ρ g h$ model water is treated as incompressible, so $ρ$ stays constant; the pressure rises because $h$ grows, not $ρ$ . See Pressure in fluids.

Why questions

Recall Explain the mechanism, not just the label

Why do we divide mass by volume instead of, say, subtracting? ::: We want "how concentrated" — a rate of mass per unit space. Division answers "how much per one unit of volume"; subtraction would mix incompatible quantities and depend on the total size. Why does specific gravity immediately tell you float-or-sink, while raw density in kg/m³ does not at a glance? ::: SG is already measured against water, so the threshold is simply $1$ : above sinks, below floats. Raw density forces you to first recall water $= 1000$ and compare. Why is density called an intrinsic property while mass and volume are not? ::: Mass and volume scale with how much stuff you take, but their ratio is fixed for a given material at fixed conditions — it fingerprints the substance itself. Why must the mixing formula assume "volumes add"? ::: Some mixtures shrink (water + alcohol pack tighter), so $V \neq = V_{1} + V_{2}$ ; the simple formula only holds when no such contraction occurs. Why does a hydrometer float higher in a denser liquid? ::: A denser liquid supplies more buoyant force per unit submerged volume, so less of the hydrometer needs to sink to balance its weight, and it rides higher — reading a larger SG. See Relative density measurement (hydrometer). Why can two objects of identical density have completely different masses? ::: Density fixes only the mass-to-volume ratio; a large and a small chunk of the same material share $ρ$ but hold different amounts of material, hence different masses.

Edge cases

Recall Boundary and degenerate scenarios

What is the "density" of a perfect vacuum, and does the formula survive? ::: With no mass in the volume, $m = 0$ , so $ρ = 0/ V = 0$ . The formula is fine as long as $V \neq = 0$ . Can $ρ = m / V$ be evaluated for zero volume? ::: No — division by $V = 0$ is undefined, matching the physical fact that "no space" cannot hold matter to speak of a concentration. Two liquids that don't mix (oil on water) — is there a single density for the pair? ::: Not a single uniform one; each layer keeps its own $ρ$ . A "mixture density" only makes sense if you blend them into one uniform body. If you compress a gas to half its volume (mass unchanged), what happens to its density? ::: It doubles, since $ρ = m / V$ and $V$ halved — gases are why the "incompressible" assumption of Continuity equation and $P = ρ g h$ is reserved for liquids. Harmonic mean of equal masses when $ρ_{1} = ρ_{2} = ρ$ : what do you get? ::: $\frac{2 ρ \cdot ρ}{ρ + ρ} = ρ$ — both mean formulas collapse to the same value when the ingredients are identical, a good sanity check. A block with SG exactly $0$ (imaginary massless block) placed on water — where does it sit? ::: It rests entirely on the surface with zero submersion, since $V_{sub} / V = SG = 0$ ; no buoyancy is needed to support zero weight. Does heating a solid change its density, and which way? ::: Usually it lowers density: heat expands the volume while mass stays fixed, so $ρ = m / V$ drops. (Water between $0$ and $4^{\circ} C$ is the famous exception.)

Connections

Buoyancy and Archimedes' principle — every float/sink verdict here rests on it.
Pressure in fluids — the $ρ$ in $P = ρ g h$ is why "deep = denser" is a trap.
Relative density measurement (hydrometer) — SG-reading instrument behind the hydrometer question.
Continuity equation — incompressibility keeps $ρ$ constant.
Pascal's law