How much mass is in each cubic centimeter
A small block of metal has a volume of 4 cm³ and a mass of 12 g. Picture the block cut into 4 cubes, each 1 cm long, 1 cm wide and 1 cm tall, so each cube is 1 cm³. The metal is the same all the way through, so the 12 g are shared equally among the 4 cubes: each cubic centimeter holds 12 ÷ 4 = 3 grams.
The mass in each cubic centimeter is called the density of the material. This metal has a density of 3 g/cm³, read "3 grams per cubic centimeter".
12 g shared over 4 cubic centimeters: 3 g in each one.
Density = mass ÷ volume
To find a density, divide the mass by the volume: density = mass ÷ volume. This is the same kind of calculation as speed, which is distance ÷ time. Both share an amount out over each single unit: speed over each hour or second, density over each cubic centimeter.
Density belongs to the material, not to the size of the block. A bigger block of the same metal, 8 cm³, has a mass of 24 g, and 24 ÷ 8 = 3, so its density is 3 g/cm³ again. The bigger block is heavier, but each of its cubic centimeters holds the same 3 g.
A block of the same metal twice as big: 24 g over 8 cm³, still 3 g in each cubic centimeter.
A different material packs a different mass into each cubic centimeter. A block of plastic the same size as the first, 4 cm³, might have a mass of only 4 g. Its density is 4 ÷ 4 = 1 gram per cubic centimeter, a third of the metal's.
The same 4 cm³ of plastic holds only 4 g: 1 g in each cubic centimeter.
Finding the mass or the volume
Once you know the density, it tells you the rest. Every cubic centimeter of the metal holds 3 g, so 40 cm³ of it has a mass of 3 × 40 = 120 g. That is mass = density × volume.
It works the other way too. To find how much space 60 g of the metal fills, ask how many lots of 3 g make 60 g: 60 ÷ 3 = 20, so the volume is 20 cm³. That is volume = mass ÷ density.
Each cubic centimeter adds another 3 g, so the mass is always 3 times the volume.
The usual mistakes
Multiplying the mass by the volume to find the density. Density shares the mass out over the volume, so it divides: for 12 g in 4 cm³, 12 ÷ 4 = 3, so the density is 3 g/cm³, not 12 × 4 = 48.
Adding the density to the volume to find the mass. Each of the 40 cm³ holds 3 g, so the mass is 40 × 3 = 120 g, not 40 + 3 = 43 g.
Thinking that the heavier object must be the denser one. The 8 cm³ block of metal is heavier than the 4 cm³ block, but both have a density of 3 g/cm³. To compare densities, compare the mass in one cubic centimeter of each.
Which object floats
An object floats in a liquid when its density is less than the liquid's density, and sinks when its density is greater. In the next problem, the volume of the oil in a tank comes from Volume of a Cuboid, and then its density gives its mass. After that, two objects are compared by their densities, not by their masses.
Worked example: A Tank of Cooking Oil: Its Mass, and Which of Two Objects Floats in It
Question In a science lab, a tank 50 cm long, 40 cm wide and 45 cm tall holds cooking oil to a depth of 30 cm. The density of the oil is 0.9 g/cm3. An object floats in a liquid when its density is less than the liquid's, and sinks when its density is greater. (a) What is the mass of the oil, in kilograms? (b) A pine block measuring 20 cm by 10 cm by 10 cm has a mass of 1200 g, and a rubber stopper with a volume of 50 cm3 has a mass of 60 g. Find the density of each. Which of them floats in the oil?
1.The oil fills 50 × 40 × 30 = 60000 cm3, which is 6 blocks of 10000 cm3.
The oil fills 50 × 40 × 30 = 60000 cm3: 6 blocks of 10000 cm3. 2.Each cubic centimeter has a mass of 0.9 g, so one block has a mass of 10000 × 0.9 = 9000 g.
Each cm3 has a mass of 0.9 g, so a block has a mass of 10000 × 0.9 = 9000 g. 3.Six blocks: 6 × 9000 = 54000 g.
Six blocks: 6 × 9000 = 54000 g. 4.(a) 54000 g = 54 kg.
(a) 54000 g is 54 kg. 5.The pine block has a volume of 20 × 10 × 10 = 2000 cm3, so each cubic centimeter of it has a mass of 1200 ÷ 2000 = 0.6 g.
The pine block has 1200 ÷ 2000 = 0.6 g in each cm3, less than the oil’s 0.9 g. 6.Each cubic centimeter of the stopper has a mass of 60 ÷ 50 = 1.2 g.
The stopper has 60 ÷ 50 = 1.2 g in each cm3, more than the oil’s 0.9 g. 7.(b) 0.6 g is less than the oil's 0.9 g, so the pine block floats; 1.2 g is more, so the stopper sinks, even though it is the lighter of the two objects.
(b) The pine block floats and the stopper sinks, though the stopper is the lighter of the two.
Answer: (a) 54 kg; (b) the block's density is 0.6 g/cm3 and the stopper's is 1.2 g/cm3, so the pine block floats and the stopper sinks
Common mistakes
- Deciding by mass that the light stopper floats and the heavy block sinks: what decides it is the mass in each cubic centimeter, the density, not the whole mass.
- Using the tank's height of 45 cm for the volume of the oil: the oil reaches a depth of only 30 cm.