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Density is mass divided by volume; it belongs to the material, not the piece, and decides what floats in water.
Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.
By the end of this lesson you will be able to find a density, a mass or a volume from the other two, and use density to predict floating.
You can measure mass with a balance and volume with a ruler or a measuring cup. You know a cubic centimeter and a milliliter are the same size. This lesson combines mass and volume into a single number that tells materials apart.
| Term | What it means |
|---|---|
| Density | Mass per unit volume, $\rho = m/V$. |
| Mass | How much matter an object has, in grams or kilograms. |
| Volume | How much space an object takes up, in cm³, mL or m³. |
| Grams per cubic centimeter | A unit of density, g/cm³: grams in each cubic centimeter. |
| Water displacement | Finding a volume by the rise in water level when an object is sunk. |
| Float | To rest at the surface, which happens when an object is less dense than the liquid. |
Density measures how tightly mass is packed into a material:
$$\rho = \dfrac{m}{V}.$$
Rearranged, $m = \rho V$ finds a mass and $V = m/\rho$ finds a volume.
Another way: picture
Picture a shoebox full of feathers and the same shoebox full of sand. They take up the same space, but the sand box is far heavier. The sand packs much more mass into every cubic centimeter: it has a higher density.
Another way: steps
Grams per cubic centimeter means how many grams sit in each cubic centimeter. If $40$ cubic centimeters of a metal have a mass of $108$ grams, each cubic centimeter carries $108 \div 40 = 2.7$ grams.
Saying the unit aloud keeps the division the right way around. Grams come first, so the mass goes on top. Dividing the other way would give cubic centimeters per gram, a different quantity.
Cut an aluminum block in half. Each half has half the mass and half the volume, so the density, mass over volume, stays exactly the same. A pin and a steel ship's hull have the same density because they are made of the same steel.
That is what makes density useful: it identifies a material. Measuring a sample's density and comparing it with a table can reveal whether a ring is gold or a coin is counterfeit.
A large log can be much heavier than a small steel nail, yet the log floats and the nail sinks. The log has more mass but spread through a much larger volume, so its density is lower.
Heaviness depends on how much of the material there is. Density depends only on what the material is. Keeping these two ideas apart is the key to this lesson.
For a box-shaped object, volume is length times width times height. A block $5$ cm by $4$ cm by $2$ cm has a volume of $40$ cm³.
For an irregular object like a stone, use water displacement. Fill a graduated cylinder partly with water, note the level, lower the stone in, and note the new level. The rise in milliliters equals the stone's volume in cubic centimeters.
Water has a density of $1$ g/cm³. An object less dense than water floats; one denser than water sinks. Ice, at about $0.92$ g/cm³, floats; iron, at $7.9$ g/cm³, sinks.
The same rule works for other liquids. Oil, less dense than water, floats on top of it. Pour honey, water and oil into a glass and they settle into layers, densest at the bottom.
Checking an answer. Metals have densities of a few to twenty grams per cubic centimeter; woods and plastics, around one or less. A density far outside those ranges usually means the division went the wrong way.
Density is defined as mass divided by volume, so the formula is simply the definition. Because mass and volume grow together for any one material, their ratio stays fixed, which is why density describes the material.
Water displacement works because a submerged object pushes aside exactly its own volume of water. The water has nowhere to go but up, so the rise in level measures the object's volume.
Scientists often use kilograms per cubic meter. Water's density is $1000$ kg/m³, the same as $1$ g/cm³. To convert from g/cm³ to kg/m³, multiply by one thousand.
A cubic meter is a big box, a meter on each side. A cubic meter of water has a mass of a thousand kilograms, a metric ton. A cubic meter of air has a mass of only about $1.2$ kg.
A famous story says the Greek scientist Archimedes was asked to check whether a king's crown was pure gold. He could not melt it down, but he could find its volume by water displacement, and its mass on a balance.
Dividing gave the crown's density. If it was less than gold's, the goldsmith had mixed in cheaper, lighter silver. Density tests like this are still used to detect fake gold today.
Salad dressing separates because oil is less dense than vinegar and floats on top. Shaking mixes them briefly, but they settle back into layers.
An egg can tell you how fresh it is. A fresh egg is dense and sinks in water. As it ages, air seeps in and its density drops, until an old egg floats.
The most common error is dividing volume by mass instead of mass by volume. Another is mixing units, such as grams with cubic meters, which gives numbers a million times off.
A third is thinking a bigger piece of the same material has a bigger density. More material means more mass and more volume, but the same density.
A few densities in grams per cubic centimeter are worth remembering: water $1$, ice $0.92$, aluminum $2.7$, iron and steel $7.9$, copper $8.9$, lead $11.3$ and gold $19.3$. Most woods fall between $0.4$ and $0.9$.
Knowing these makes it easy to judge an answer. A metal block with a calculated density of $0.02$ g/cm³, or a wooden block of $50$ g/cm³, signals a mistake.
If you know a material's density and an object's volume, multiplying gives its mass. An aluminum block of $100$ cm³ has a mass of $2.7 \times 100 = 270$ g.
Engineers use this to find the mass of steel beams, concrete floors and fuel in a tank from their dimensions, without ever putting them on a scale.
Gases are far less dense than liquids and solids, because their molecules are spread far apart. Air is about eight hundred times less dense than water.
Hot air is a little less dense than cool air, which is why hot-air balloons rise. Helium is much less dense than air, which is why a helium balloon floats up if you let go of its string.
Most substances are densest as solids, so a solid piece sinks in its own liquid. Water is unusual. When it freezes, its molecules lock into an open pattern that takes up more space, so ice is about eight percent less dense than liquid water and floats.
That odd fact matters for life. Lakes in Minnesota and Michigan freeze from the top down, and the floating ice insulates the water below, so fish survive the winter in liquid water under the ice. If ice sank, lakes would fill with ice from the bottom up.
Most materials expand a little when heated, spreading the same mass over more volume, so their density falls. That is why warm water rises in a pot heated from below and cool water sinks, setting up the circulating currents you can see when you heat soup.
The same idea drives the weather. Warm air near the ground is less dense than the cooler air above, so it rises, carrying moisture up to form clouds on a summer afternoon.
Because each material has its own density, measuring density is one of the simplest ways to identify an unknown sample. Geologists use it to tell minerals apart, recyclers use it to sort plastics in water tanks, and museum conservators use it to check whether an old coin is genuine without scratching it.
Density depends on two measurements, so errors in either one matter. Read a graduated cylinder at eye level, at the bottom of the curved water surface, and make sure no air bubbles cling to a sunken stone.
The United States Bullion Depository at Fort Knox, Kentucky, stores thousands of gold bars, each with a mass of about twelve and a half kilograms. Gold is so dense, about nineteen grams per cubic centimeter, that a bar that heavy is only about the size of a thick paperback book.
Density is also how gold is checked. Tungsten has almost the same density as gold, which has made it a favorite of counterfeiters who coat tungsten with a thin layer of gold. Assayers measure a bar's mass and volume very precisely and compare the density with gold's; they also use sound waves and other tests, because a close match in density alone is not quite enough to be sure.
When oil spills from a tanker or a well, as in the Gulf of Mexico in 2010, most of it floats. Crude oil is typically about eighty-five to ninety percent as dense as seawater, so it spreads into a thin slick on the surface.
Cleanup crews use that difference in density. Floating barriers called booms hold the oil in place, and skimmers collect it from the top. Some heavy oils, denser than fresh water, can sink in rivers, which makes them much harder to recover. Knowing an oil's density helps responders decide which tools will work.
It is natural to think heavy objects are dense and light ones are not. But a large wooden log is heavy yet floats, while a tiny steel nail is light yet sinks. Density compares mass to volume, not mass alone.
A related error is to think a bigger piece of a material has a bigger density. Doubling the size doubles both mass and volume, so the density stays the same. Density is a property of the material, not of the piece.
A metal cube has a mass of $216$ g and sides of $3$ cm. Find its volume.
$V = 3 \times 3 \times 3 = 27\ \text{cm}^3$
Side cubed.
Find its density.
$\rho = \dfrac{216}{27} = 8\ \text{g/cm}^3$
Mass over volume.
Compare with a table.
$\text{steel: about } 7.9\ \text{g/cm}^3$
Close match.
Predict floating or sinking.
$8 > 1 \Rightarrow \text{sinks}$
Denser than water.
Find the density of half the cube.
$\dfrac{108}{13.5} = 8\ \text{g/cm}^3$
Unchanged.
A cylinder holds $60$ mL of water. A $75$ g stone raises it to $90$ mL. Find the stone's volume.
$V = 90 - 60 = 30\ \text{cm}^3$
Rise in level.
Find its density.
$\rho = \dfrac{75}{30} = 2.5\ \text{g/cm}^3$
Mass over volume.
Compare with common rocks.
$\text{granite: about } 2.7\ \text{g/cm}^3$
Typical rock.
Convert to kg/m³.
$2.5 \times 1000 = 2500\ \text{kg/m}^3$
Times a thousand.
Find the mass of a $1000$ cm³ boulder of it.
$m = 2.5 \times 1000 = 2500\ \text{g}$
$m = \rho V$.
Convert that mass to kilograms.
$2.5\ \text{kg}$
Divide by a thousand.
A log is $200$ cm long, $30$ cm wide and $30$ cm high, with a mass of $108$ kg. Find its volume.
$V = 200 \times 30 \times 30 = 180000\ \text{cm}^3$
A box shape.
Convert the mass to grams.
$108000\ \text{g}$
Times a thousand.
Find its density.
$\rho = \dfrac{108000}{180000} = 0.6\ \text{g/cm}^3$
Mass over volume.
Predict whether it floats.
$0.6 < 1 \Rightarrow \text{floats}$
Less dense than water.
A steel nail has a mass of $5$ g and a volume of $0.63$ cm³. Find its density.
$\rho = \dfrac{5}{0.63} = 7.9\ \text{g/cm}^3$
Mass over volume.
Compare the two objects.
$\text{log heavier, nail denser}$
Different questions.
Predict what the nail does.
$7.9 > 1 \Rightarrow \text{sinks}$
Denser than water.
Write the formula.
$\rho = \dfrac{m}{V}$
Mass over volume.
Substitute the values.
$\rho = \dfrac{135}{50}$
Grams over cm³.
Evaluate the density.
A small block has a mass of $26.7$ g and a volume of $3$ cm³. What is its density, in g/cm³?
Complete the worked solution: a graduated cylinder holds $50$ mL of water. When a $54$ g stone is lowered in, the level rises to $70$ mL. Find the stone's volume in cm³, its density in g/cm³, and the mass in g of a stone of the same material twice as big.
Find the stone's volume.
$V = V_{\text{after}} - V_{\text{before}} =$ v
The water it pushed up.
Find the density.
$\rho = \dfrac{m}{V} =$ d
Mass over volume.
Find the bigger stone's mass.
$m_2 = \rho \times 2V =$ t
Same density.
Explain the method.
$\text{works for any shape}$
No ruler needed.
Match each idea to its statement.
| mass divided by volume | less dense than water | the same density as the whole block | 2.7 grams in each cubic centimeter | |
|---|---|---|---|---|
| density | ||||
| an object that floats in water | ||||
| half of a block | ||||
| a density of 2.7 g/cm³ |
A rectangular block measures $10$ cm by $5$ cm by $2$ cm and has a mass of $790$ g. Fill in its volume in cm³, its density in g/cm³, and the mass in g of a block of the same material twice as long.
| value | |
|---|---|
| volume (cm³) | |
| density (g/cm³) | |
| mass of a block twice as long (g) |
The density of vegetable oil is $0.8$ g/cm³. Write the mass, in g, of a piece of vegetable oil as a function of its volume $V$ in cubic centimeters.
Answer:
a gold ring has a mass of $12$ g, and its material has a density of $19.3$ g/cm³. What is its volume, in cm³?
Answer: cm³
a sixteen-pound bowling ball has a mass of about $7260$ g, and its average density is about $1.33$ g/cm³. What is its volume, in cm³?
Answer: cm³
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
The density of aluminum is $2.7$ g/cm³. Write the mass, in g, of a piece of aluminum as a function of its volume $V$ in cubic centimeters.
Answer:
You can use density. Explain to someone why a heavy log floats while a light steel nail sinks.
29. Your turn: a block has a mass of $135$ g and a volume of $50$ cm³. What is its density?, step 3
$\rho = 2.7\ \text{g/cm}^3$
Aluminum.