This is middle school physics' first "combination" measurement: no single reading gives you the answer — you must weigh once, measure once, then do one division. It is also a fixture on lab exams, where the points lost are almost never in the arithmetic but in the order of operations, the readings, and the little procedural details that look like they shouldn't matter. This guide walks through the real procedure, then uses an interactive simulation to pull the variables apart.

Safety first (read this section before you start)

  • A glass graduated cylinder is tall, narrow and top-heavy. Keep it on a flat surface well back from the table edge.
  • When putting the rock into the cylinder, tilt the cylinder and let the rock slide slowly down the inside wall. Never drop it straight in from the top — a rock hitting the glass bottom can punch right through it.
  • Wash and dry the rock beforehand and check it for sharp edges; blunt any dangerous corners with sandpaper first.
  • On a pan balance, always handle the standard masses with forceps, never with bare fingers (sweat corrodes them and changes their mass), and never roll them around on the tabletop.
  • Do not put wet objects directly on the balance pan, and never weigh chemicals straight on the pan.
  • When you finish, return the masses to their box in size order, slide the rider back to zero, and empty the cylinder and stand it upside down to dry.

Materials and equipment

  • A two-pan balance (with its box of masses and forceps), or an electronic balance;
  • A graduated cylinder, 50 mL or 100 mL — big enough to take the rock with water level to spare;
  • Two or three small rocks (small enough to fit the cylinder, each displacing at least 10 mL), ideally of different kinds;
  • A length of thin string, plus a thin wire or a pin (for lowering the rock and for the pin-press method);
  • Fresh water, a dry cloth, and paper and pen for recording.
  • Optional: a small block of wood or wax, to try the "what if it floats" case.

Step 1: level the balance, then weigh

If the balance isn't leveled first, none of the later data counts. Set the balance on a level tabletop, slide the rider to the zero mark at the far left of its scale, and watch the pointer: if it swings left, turn the leveling nut to the right; if it swings right, turn the nut left — until the pointer rests at the center of the dial, or swings equally far to each side. Remember the rule of thumb: turn the nut opposite to the pointer.

Once leveled, place the rock on the left pan and add masses to the right pan with the forceps, largest first: try the biggest, and if the pointer tips toward the masses, swap it for the next size down; keep going until one mass just tips it over, then trade down again. When every mass has been tried and the pointer still leans toward the rock, switch to sliding the rider until the balance levels out again.

Reading = total of the masses + the rider's scale mark. Say you used the 50 g, 20 g and 5 g masses and the rider sits at 2.4 g: the rock's mass is m = 77.4 g. Write that number down before anything else.

Why "object left, masses right": most people handle the forceps with their right hand, so the object sits more conveniently on the left. More importantly, if you swap sides and then use the rider, the reading becomes "masses minus rider" — an easy way to get it wrong. A fixed habit beats re-deriving the formula every time.

Step 2: find the volume by water displacement

Pour a suitable amount of water into the cylinder. "Suitable" has two hard requirements: the water must be able to cover the rock completely, and once the rock is in, the level must not rise past the cylinder's top graduation. Set the cylinder level, bring your eye level with the lowest point of the meniscus, and read V₁.

Tilt the cylinder slightly, lower the rock slowly to the bottom on its string, stand the cylinder upright again, and once the surface is still, read at eye level again to get V₂. The rock's volume is

V = V₂ − V₁

If V₁ = 40 mL and V₂ = 68 mL, then V = 28 mL = 28 cm³. Note that 1 mL is exactly 1 cm³ — the two units are numerically identical, so either is fine in your answer.

Swap in a block of wood or wax and it floats, its exposed part displacing no water, so the V₂ you read is too small. In that case push the whole block below the surface with a pin or thin wire before reading — the pin-press method; or tie on a sinker of known volume, submerge them together, and subtract the sinker's volume — the sinker method.

Step 3: calculate the density and check the table

Put the two numbers into the definition:

ρ = m ÷ V = 77.4 g ÷ 28 cm³ ≈ 2.76 g/cm³

Most common rocks fall between 2.5 and 3.0 g/cm³: granite about 2.6–2.7, basalt about 2.9–3.0, quartz about 2.65. A result of 2.76 says you're most likely holding an ordinary silicate rock. Measure a second rock the same way: if the two densities come out close, they are probably the same kind of rock; if they differ widely, they are different minerals. That is density's job as a substance's "ID card."

Interactive simulation: this experiment has a drag-and-play simulator in our Interactive Lab. The lab interface is currently Chinese-only — the controls are simple sliders and switches, so it is still easy to explore.

What you should observe

Compare your students' data against these benchmarks:

  • Measuring the same rock three times, masses usually agree within 0.2 g and volumes within 1 mL, so the calculated densities differ by no more than about 0.1 g/cm³.
  • Switch to a larger rock of the same kind: m and V both grow, but the ratio barely moves.
  • Take the rock out of the water and weigh it without drying it, and the mass comes out a few tenths of a gram high — pushing the calculated density up with it.
  • Leave the wood block floating and the V₂ you read is clearly too small; press it under and V₂ grows, bringing the calculated density back down to a sensible value around 0.5.
  • Reading the cylinder while looking down at the surface gives a number that is too high; looking up from below gives one too low. This error hits V₁ and V₂ in opposite directions, so the mistakes can compound.

The science

For any one substance, the ratio of mass to volume is a fixed value, and that ratio is called density, written ρ = m ÷ V. It expresses "how much one unit of volume of this substance weighs," which makes it a characteristic property of the substance: snap the rock in half and each half loses half its mass and half its volume at once — the ratio doesn't budge. This is also why you can't simply say "iron is heavier than cotton"; the comparison only works once you add "for the same volume."

Water displacement measures volume thanks to one simple fact: a fully submerged object displaces a volume of water equal to its own volume. "Fully submerged" is therefore the precondition of this step — let any part poke out and the measured volume is too small. And since V sits in the denominator of ρ = m ÷ V, a small V makes ρ come out large: the classic systematic error of this experiment.

The order matters too. Always weigh first, measure volume second: do it the other way around and the rock comes out of the water with a film of water clinging to it, gets weighed water and all, m reads high, and ρ follows. Designs where "the order changes the conclusion" are exactly what exam questions love to probe. Incidentally, for measuring a liquid's density the order flips: weigh "beaker + liquid" first, pour the liquid into the cylinder and read its volume, then weigh the now-empty beaker and get the poured liquid's mass by subtraction — sidestepping the error from liquid left clinging to the beaker's walls.

The balance's reading rule deserves its own line: mass = total of the standard masses + the rider reading. The rider is really a "small mass you can slide continuously," filling in the remainder below the smallest standard mass. Forgetting to read the rider is this experiment's second most common way to lose points.

Notes for teachers and parents

  • Time and grouping: two class periods, groups of four; share the balance and cylinder, taking turns.
  • Say this before hands touch anything: the graduated cylinder is glassware — handle it gently.

The step most often skipped. Have every group measure the same rock once and write the whole class's numbers on the board — the spread in the data is the best possible material for teaching error. Don't rush to correct a bad reading; first ask, "does this step push the result up or down?"

How to tell they truly understand. A student who can weigh and run the displacement measurement independently, read the meniscus correctly, and state which direction a given mistake pushes the result, has met the goal.

Take it further, and common misconceptions

  • Misconception 1: "A denser object must be heavier." No — heaviness is about mass; density only settles which is heavier when the volumes are equal.
  • Misconception 2: "ρ is proportional to m and inversely proportional to V." That reads the definition backwards. ρ is the ratio of m to V and does not change when either one changes alone.
  • Misconception 3: "You can ignore the rider." Skipping the rider silently drops several grams and skews the result.
  • Misconception 4: "Read the cylinder at the highest point of the liquid." Read the lowest point of the meniscus, with your line of sight horizontal.
  • Misconception 5: "A substance's density never changes." True only at a fixed state and temperature. Water expands when it freezes, its density dropping from 1.0 to about 0.9 g/cm³ — which is why ice floats, and why pipes burst in winter.
  • Advanced variant: if the rock won't fit in the cylinder, use an overflow can — fill it to the spout, lower in the rock, catch the overflow and measure that water in the cylinder. The principle is identical.
  • To see how density decides floating and sinking, read on: why an egg floats in salt water.

Only when a student can look at any measurement and explain "this step pushes the result up, that one pushes it down" has this experiment truly been understood.