Press Next (or Play) to walk through each story one small step at a time. The numbers come last. After every two stories there is one quick question; if it feels shaky, press Show me another example for one more story. Tap the small round play button next to a caption to hear it read aloud.
1. Is this bar really gold?
Read the story as text
You buy a "pure gold" bar online. It looks gold and it feels heavy. But a cheap metal painted gold also looks gold.
You put the bar on a kitchen scale: 500 g. Then you drop it into a measuring cup of water. The water level goes up by 40 cm³.
You cannot see inside the bar. But you can measure two numbers from the outside: how much mass it has, and how much space it takes up.
The question: how can mass and volume together tell you what something is made of, and why do some things float on others?
2. Same size, different stuff
Quick check
Two blocks are exactly the same size. Block A has twice the mass of block B. How do their densities compare?
Another example: Squashing bread
3. Float or sink?
4. Oil floats up
Quick check
A toy with a density of 800 kg/m³ is dropped into water (1000 kg/m³). What happens?
Another example: The floating egg
5. The hot air balloon
6. The big log and the tiny pebble
Quick check
A huge log floats on a lake, but a tiny pebble sinks. Why?
Another example: A bucket of water, a bucket of sand
7. Check yourself
Think of your answer first, then tap to see it.
a) Maya cuts her bar into two equal halves. What is the density of each half?
Show answer
The same as the whole bar, 12.5 g/cm³. Each half has half the mass and half the volume, and half ÷ half is the same ratio. Density belongs to the material, not to the size of the piece.
b) A plastic toy has a density of 0.9 g/cm³. Does it float or sink in water (1.0 g/cm³)? About how much of it is under water?
Show answer
It floats, because it is less dense than water. About 0.9 ÷ 1.0 = 90% of it sits under the surface, like the ice cube.
c) A stone has a mass of 270 g. Dropped into a measuring cup, it makes the water rise by 100 cm³. What is its density?
Show answer
ρ = m ÷ V = 270 g ÷ 100 cm³ = 2.7 g/cm³, which is 2700 kg/m³ (multiply by 1000). It is denser than water, so it sinks.
Already know this?
Three quick questions. Get all three right on the first try and you can skip ahead to the simulation. Not sure? No problem: just read on.
1) A 2 kg steel ball and a 1 kg steel ball are made of the same steel. Compare their densities.
2) A 200 g object takes up 250 cm³. What is its density?
3) An ice cube floats in a glass of water. Which statement is true?
The idea in plain words
What is a fluid? Matter is made of tiny particles (atoms and molecules). They pull and push on each other.
In a solid the particles are locked in place. So a solid keeps its shape.
In a liquid the particles touch but slide past each other. So a liquid takes the shape of its container.
In a gas the particles are far apart and fly around. So a gas fills its whole container.
A fluid is anything with no fixed shape: liquids AND gases.
Now hold two blocks of the same size. One is wood. One is lead. The lead block is far heavier. Same space, but much more mass packed in.
That packing is density. Density tells you how much mass is packed into each bit of space.
In words: density equals mass divided by volume:
ρ = m / V
ρ (the Greek letter "rho") is density in kg/m³, m is mass in kg, V is volume in m³. Density belongs to the material,
not the object. Cut a steel bar in half: half the mass, half the volume, the same density. That is why density can identify a material.
Trap: changing cm³ to m³ by dividing by 100 (or 1000). Instead: 1 cm = 10⁻² m, so 1 cm³ = (10⁻²)³ = 10⁻⁶ m³. 40 cm³ = 4.0 × 10⁻⁵ m³. Also 1 g/cm³ = 1000 kg/m³.
Handy numbers: water is 1000 kg/m³ (= 1.0 g/cm³). Ice 920, oil about 900, aluminum 2700, iron 7900, lead 11 300, gold 19 300 kg/m³. Air is only about 1.2 kg/m³.
Trap: "it is heavy, so it is dense" (a big log is heavy but floats). Instead: density is mass per volume, ρ = m/V. Compare m/V, not mass alone: a 50 kg log (ρ ≈ 600) is less dense than a 10 g iron nail (ρ = 7900).
Ideal fluid. On the AP exam every fluid is ideal unless the problem says otherwise. That means two things:
it is incompressible (squeezing it does not change its volume, so its density stays the same everywhere) and it has
no viscosity (no internal "stickiness" or friction as it flows). Real water is very close to incompressible. Gases are not, but AP Physics 1 does not ask you to squeeze gases.
Worked: is the bar gold?
Read the steps as text
Write down what we measured: m = 500 g = 0.500 kg, V = 40 cm³. Why: always list the knowns with units first.
Change the volume to m³: 40 cm³ × 10−6 m³/cm³ = 4.0 × 10−5 m³. Why: 1 cm = 10−2 m, so 1 cm³ = (10−2)³ = 10−6 m³.
Use ρ = m / V = 0.500 kg ÷ 4.0 × 10−5 m³ = 12 500 kg/m³. Why: density is mass per volume.
Compare with gold: 19 300 kg/m³. Why: a material has one density; if it does not match, it is not that material.
Check with a size estimate: real gold of 500 g would have V = m / ρ = 0.500 / 19 300 = 2.6 × 10−5 m³ = 26 cm³. Why: our bar takes 40 cm³, too much space for gold. It is a fake (maybe a lead-like alloy inside).
Lab 1: measure density like a scientist
Pick the object's mass and volume. A thin rod lowers it into a graduated cylinder (inside area 50 cm²) until it rests on the bottom,
so it goes fully under even if it would float. Watch the water reading. The rise in the reading is the object's volume. Lab units here are grams and cm³; the note converts to kg/m³.
Mass vs volume
Each material is a straight line through the origin. Its slope is its density.
Lab 2: the density column
A tall jar holds three liquids that do not mix: oil on top, water in the middle, syrup at the bottom. Denser liquids sink below less dense ones.
Drop a 10 cm cube of any density at the top surface and see where it stops. The liquids slow the cube down; the sim uses a simple drag (damping) term for that, so the cube settles instead of bobbing forever.
Try: cube 950 kg/m³ (between oil and water), cube 1100 (between water and syrup), then make the oil denser than the cube.
Examples
Example 1: a bottle of olive oil basic
A 2.0 L bottle holds 1.84 kg of olive oil. What is the density of the oil?
Show solutionRead the steps as text
V = 2.0 L = 2.0 × 10−3 m³ (1 L = 10−3 m³). ρ = m / V = 1.84 / 2.0 × 10−3 = 920 kg/m³. Less than water, so oil floats on water.
Example 2: cutting an aluminum block medium
An aluminum block (ρ = 2700 kg/m³) is 5.0 cm × 4.0 cm × 2.0 cm. (a) What is its mass? (b) It is cut in half. What are the mass and density of one half?
Show solutionRead the steps as text
(a) V = 5.0 × 4.0 × 2.0 = 40 cm³ = 4.0 × 10−5 m³. m = ρV = 2700 × 4.0 × 10−5 = 0.108 kg (108 g).
(b) Half the mass, 0.054 kg, and half the volume, 2.0 × 10−5 m³. Density = 0.054 / 2.0 × 10−5 = 2700 kg/m³, unchanged. Density is a property of the material.
Example 3: density from a graph AP
A student measures four samples of one unknown metal:
Volume (cm³)
10
20
30
40
Mass (g)
27
53
82
108
(a) What should she graph to find the density? (b) Find the density in kg/m³ and name a likely metal.
Show solutionRead the steps as text
(a) Mass on the vertical axis, volume on the horizontal axis. Since m = ρV, the graph is a straight line through the origin and its slope is ρ.
(b) Use the best-fit line, not one data point. Slope ≈ (108 − 27) g ÷ (40 − 10) cm³ = 81 / 30 = 2.7 g/cm³ = 2700 kg/m³, aluminum.
Using many points and a slope averages out small errors in each measurement.
Example 4: mixing two liquids AP
Water (1000 kg/m³) and alcohol (790 kg/m³) are mixed. Treat the volumes as simply adding. Find the density of the mix if you use (a) equal volumes, 1.0 L each; (b) equal masses, 1.0 kg each.
Show solutionRead the steps as text
(a) Masses: 1.00 kg + 0.79 kg = 1.79 kg in 2.0 × 10−3 m³. ρ = 1.79 / 2.0 × 10−3 = 895 kg/m³ (the plain average).
(b) Volumes: 1.0/1000 = 1.00 × 10−3 m³ and 1.0/790 = 1.27 × 10−3 m³. Total 2.27 × 10−3 m³ for 2.0 kg. ρ = 2.0 / 2.27 × 10−3 ≈ 880 kg/m³.
Not the plain average: there is more alcohol by volume, so the mix is closer to alcohol's density. Always go back to total mass ÷ total volume.
Practice (AP style)
Which statement best describes a fluid?
Show answer
(B). Liquids and gases both flow and take the shape of their container. (A) leaves out gases. (C) mercury is a fluid denser than water. (D) particles in a fluid do interact; those interactions cause pressure (topic 8.2).
A uniform block of density ρ is cut into three pieces of different sizes. What is the density of the smallest piece?
Show answer
(B). Mass and volume shrink by the same factor, so m/V does not change. Density is a property of the material, not of the piece.
Cubes A and B have the same mass. The side of cube B is twice the side of cube A. How do their densities compare?
Show answer
(C). Volume goes with side cubed: VB = 2³ VA = 8VA. Same mass, 8 times the volume, so ρB = ρA/8. (A) forgets to cube; (B) squares instead of cubing.
250 cm³ of a liquid has a mass of 0.30 kg. What is its density?
A solid sphere of radius 0.10 m has a mass of 3.0 kg. Will it float in water?
Show answer
(B). V = (4/3)π(0.10)³ = 4.19 × 10−3 m³. ρ = 3.0 / 4.19 × 10−3 ≈ 716 kg/m³ < 1000 kg/m³, so it floats. (A) is a factor-of-10 slip; (C) comes from a wrong volume formula; (D) mass alone never decides floating (ships float).
Which explains why squeezing a sealed syringe full of air is easy, but a sealed syringe full of water barely moves?
Show answer
(B). Compressibility comes from the space between particles. That is why we model liquids as incompressible (ideal). Viscosity (C) is about flow, not squeezing.
Free response (translation between representations). On a graph of mass (vertical) vs volume (horizontal), liquid A's line passes through (50 cm³, 60 g) and liquid B's line passes through (50 cm³, 40 g). Both lines start at the origin.
(a) Which line is steeper, and what does that mean? (b) The liquids do not mix. Which one ends up on top in a jar? (c) What is the mass of 200 cm³ of liquid B?
(c) m = ρV = 0.80 g/cm³ × 200 cm³ = 160 g = 0.16 kg (or read the line: 4 × 40 g).
Free response (experimental design). You have a balance, a graduated cylinder, water, and several irregular rocks cut from the same stone. Describe how to find the stone's density as accurately as you can.
Show answer
1. Measure each rock's mass m on the balance. 2. Put water in the cylinder and read the level V1. 3. Lower one rock in gently so it is fully under water (no splashing, no air bubbles); read V2. The rock's volume is V2 − V1.
4. Repeat for every rock. 5. Graph m (vertical) against V (horizontal) and draw the best-fit straight line; its slope is the density. Using many rocks and the slope reduces random error. Read the cylinder at eye level at the bottom of the curved surface.
Free response (Mathematical routines). A solid aluminum cylinder (ρ = 2700 kg/m³) has radius r and height h.
(a) Derive an expression for its mass in terms of ρ, r and h.
(b) Calculate the mass for r = 2.0 cm and h = 10 cm.
(c) A second aluminum cylinder has twice the radius and twice the height. How do its mass and its density compare with the first?
Show answer
(a) V = πr²h, so m = ρV = ρπr²h.
(b) r = 0.020 m, h = 0.10 m. V = π × 0.020² × 0.10 = 1.26 × 10⁻⁴ m³. m = 2700 × 1.26 × 10⁻⁴ ≈ 0.34 kg.
(c) Mass: 2² × 2 = 8 times bigger (about 2.7 kg). Density: the same, 2700 kg/m³, because it is the same material.
Point guide (4 points): 1 for m = ρπr²h; 1 for converting cm to m; 1 for 0.34 kg; 1 for ×8 mass and same density.
Free response (Qualitative / quantitative translation). A student says: "My 2.0 kg block of wood is denser than this 0.50 kg block of aluminum, because it is heavier."
(a) Explain in words what is wrong and what you would need to know to compare densities.
(b) The wood block's volume is 3.3 × 10⁻³ m³ and the aluminum block's is 1.85 × 10⁻⁴ m³. Calculate both densities.
(c) A 1.0 L bottle is half filled with water (1000 kg/m³) and half with oil (900 kg/m³), by volume. What is the average density of the contents?
Show answer
(a) Density is mass per volume, not mass. A heavy object can be big and light for its size. You need each block's volume as well as its mass.
(b) Wood: 2.0 / 3.3 × 10⁻³ ≈ 610 kg/m³. Aluminum: 0.50 / 1.85 × 10⁻⁴ ≈ 2700 kg/m³. The aluminum is about 4.5 times denser.
(c) Mass = 1000 × 0.00050 + 900 × 0.00050 = 0.95 kg in 0.0010 m³, so ρ = 950 kg/m³ (equal volumes, so it is the plain average).
Point guide (4 points): 1 for "mass per volume, need V"; 1 for each density (2); 1 for 950 kg/m³ with work.
Common mistakes
The mistake: converting cm³ to m³ by multiplying by 10−2.
Why it is wrong: a volume has three lengths. 1 cm³ = (10−2 m)³ = 10−6 m³. Likewise 1 L = 1000 cm³ = 10−3 m³.
How to spot it: a density of a solid or liquid should be between about 100 and 20 000 kg/m³. Water must come out 1000 kg/m³.
The mistake: thinking a bigger or heavier object is denser.
Why it is wrong: density is mass per volume. A huge log has a big mass but is less dense than a small pebble.
How to spot it: if your reasoning uses only mass or only size, you have not found density. You need both.
The mistake: averaging two densities to get the density of a mixture.
Why it is wrong: that only works for equal volumes. In general ρmix = (total mass) ÷ (total volume).
How to spot it: if the problem gives equal masses (or any unequal amounts), do the total mass ÷ total volume calculation.
The mistake: saying "air is not a fluid" or "only liquids are fluids".
Why it is wrong: a fluid is anything that flows and has no fixed shape. Gases flow too (wind is flowing air), and air pushes with pressure and buoyant force.
How to spot it: ask "does it keep its own shape?" If not, it is a fluid.
The mistake: "heavy things sink, light things float".
Why it is wrong: what matters is the object's density compared with the fluid's density (topic 8.3). A 100 000 tonne ship floats; a 1 g grain of sand sinks.
How to spot it: compare ρobject with ρfluid, never mass with mass.