Exercises: Density

An eraser has a mass of $30\ \text{g}$ and occupies a volume of $15\ \text{cm}^3$. Calculate its density in $\text{g/cm}^3$ (remember: $d = m/V$).

A block of pine wood weighs $120\ \text{g}$ and has a volume of $200\ \text{cm}^3$. What is its density? Express the result in $\text{g/cm}^3$.

An iron nut has a mass of $79\ \text{g}$ and a volume of $10\ \text{cm}^3$.

1. a) Calculate its density.
2. b) State whether it is greater or less than the density of water, which is $1\ \text{g/cm}^3$.

In the kitchen, there is a bottle containing $500\ \text{cm}^3$ of oil. The density of oil is $0,9\ \text{g/cm}^3$. Calculate the mass of the oil (hint: rearrange for $m$ in the formula $d = m/V$).

The density of aluminium is $2,7\ \text{g/cm}^3$.

1. a) Explain in your own words what this piece of data means.
2. b) What is the mass of an aluminium cube of $1\ \text{cm}^3$?
3. c) What about a block of $4\ \text{cm}^3$?

The density of cork is $0,24\ \text{g/cm}^3$ and that of water is $1\ \text{g/cm}^3$, as shown in the diagram.

1. a) What happens if you drop a cork stopper into a glass of water? Justify your answer by comparing their densities.
2. b) What if you drop a steel coin, whose density is $7,9\ \text{g/cm}^3$? Justify your answer in the same way.

An ice cube (density $0,92\ \text{g/cm}^3$) falls into a glass of water (density $1\ \text{g/cm}^3$), as shown in the diagram.

1. a) Does it float or sink? Justify your answer by comparing their densities.
2. b) Relate your answer to the fact that icebergs float in the sea.

You have made a plasticine figure and want to calculate its density in the laboratory.

1. a) Which two quantities do you need to measure?
2. b) Which instrument do you use to measure each of them?
3. c) What calculation do you perform with the two measurements to find the density?

The density of water is $1\ \text{g/cm}^3$.

1. a) Knowing that $1\ \text{g/cm}^3 = 1000\ \text{kg/m}^3$, express the density of water in $\text{kg/m}^3$, which is the International System unit.
2. b) What is the mass of $1\ \text{m}^3$ of water?

Aluminium has a density of $2,7\ \text{g/cm}^3$.

1. a) Express it in $\text{kg/m}^3$ (remember: $1\ \text{g/cm}^3 = 1000\ \text{kg/m}^3$).
2. b) Aeroplanes are built using many aluminium parts. Looking at the data, explain why it is advantageous for the material of an aeroplane to have a low density.

A gold ingot has a mass of $386\ \text{g}$ and the density of gold is $19,3\ \text{g/cm}^3$. Calculate the volume of the ingot by rearranging for $V$ in the formula $d = m/V$. Express the result in $\text{cm}^3$.

To find the density of a stone, Sara first weighs it on the balance: $54\ \text{g}$. Then she pours $60\ \text{mL}$ of water into a measuring cylinder and, on submerging the stone, the level rises to $80\ \text{mL}$, as shown in the figure.

1. a) What is the volume of the stone, in $\text{cm}^3$?
2. b) Calculate its density.
3. c) The density of granite is $2,7\ \text{g/cm}^3$. Could the stone be made of granite? Reason your answer.

A plastic ball has a mass of $150\ \text{g}$ and a volume of $200\ \text{cm}^3$.

1. a) Calculate its density.
2. b) Will it float in water, whose density is $1\ \text{g/cm}^3$? Justify your answer by comparing the densities.

In a glass, we carefully pour three liquids that do not mix with each other: honey ($1,4\ \text{g/cm}^3$), water ($1\ \text{g/cm}^3$) and oil ($0,9\ \text{g/cm}^3$), as shown in the figure.

1. a) Order the three liquids from most dense to least dense.
2. b) Predict which liquid will remain at the bottom of the glass, which in the middle and which at the top, and justify your prediction.

A raw egg sinks in a glass of tap water, but if we dissolve a lot of salt in the water, the egg ends up floating, as shown in the figure.

1. a) What happens to the density of water when we dissolve salt in it?
2. b) Explain, by comparing densities, why the egg comes to float.
3. c) Relate this to how easy it is to float in the Dead Sea, which is much saltier than tap water.

You have two cubes of the same size, with a volume of $10\ \text{cm}^3$ each: one is made of balsa wood ($0,12\ \text{g/cm}^3$) and the other of lead ($11,3\ \text{g/cm}^3$), as shown in the figure.

1. a) Calculate the mass of each cube.
2. b) How many more grams does the lead cube weigh than the balsa cube?
3. c) Explain why, occupying exactly the same space, they weigh so differently.

To check whether a pendant is made of gold, you weigh it on the balance: $96,5\ \text{g}$. Then you submerge it in a measuring cylinder and the water level rises from $50\ \text{mL}$ to $55\ \text{mL}$, as shown in the figure.

1. a) Calculate the volume of the pendant.
2. b) Calculate its density.
3. c) The density of gold is $19,3\ \text{g/cm}^3$. Is the result consistent with the pendant being made of gold?
4. d) A jeweller warns you that there is another metal with a density almost identical to that of gold. Is this test enough to be completely sure? What other characteristic property would you check?

A sheet of steel dropped into water sinks (the density of steel is $7,9\ \text{g/cm}^3$), yet a steel ship weighing thousands of tonnes floats in the sea, as shown in the figure.

1. a) Explain the role played by the air inside the hull: what can you say about the average density of the ship (including the air) compared to that of water?
2. b) Predict what happens if an impact creates a hole in the hull and water takes the place of the air. Justify your answer using the concept of average density.

You already know that ice ($0,92\ \text{g/cm}^3$) floats in water ($1\ \text{g/cm}^3$).

1. a) Predict what will happen if you drop an ice cube into a glass of oil ($0,9\ \text{g/cm}^3$): will it float or sink? Justify your answer.
2. b) After a while, the ice cube melts completely and becomes liquid water. Where will that water end up, at the top or at the bottom of the glass? Justify your answer by comparing densities.
3. c) Explain why the result in part a is different from what you observe when ice is in a glass of water.

The density of air is $1,29\ \text{kg/m}^3$ and that of helium is $0,18\ \text{kg/m}^3$, as shown in the figure.

1. a) Explain, by comparing densities, why a balloon filled with helium rises if you release it.
2. b) A balloon that you blow up by blowing does not rise: what can you say about the density of the air coming out of your lungs compared to that of the air in the room?
3. c) Hot-air balloons heat the air inside them; as the air heats up, it expands and its density decreases. Explain, using the concept of density, how they manage to rise.

You want to measure the density of a cork stopper using a balance and a measuring cylinder, but a problem arises: the cork floats and does not submerge on its own, so the water in the measuring cylinder does not show its full volume.

1. a) Describe a trick to measure the total volume of the stopper (for example, tying it to a heavy object that sinks). State which measurements you would make with the measuring cylinder.
2. b) The stopper weighs $4,8\ \text{g}$ and, measured using your trick, its volume turns out to be $20\ \text{cm}^3$. Calculate its density.
3. c) Check that the result explains why cork floats in water ($1\ \text{g/cm}^3$).

These are the densities of several materials: paraffin $0,87\ \text{g/cm}^3$; oil $0,9\ \text{g/cm}^3$; ice $0,92\ \text{g/cm}^3$; water $1\ \text{g/cm}^3$; a hard plastic $1,2\ \text{g/cm}^3$; glycerine $1,26\ \text{g/cm}^3$; iron $7,9\ \text{g/cm}^3$.

1. a) Which materials from the list float in water?
2. b) Which float in oil?
3. c) Is there any material that sinks in oil but floats in water? Which one?
4. d) And any material that floats in glycerine but sinks in water? Which one?

Justify each answer by comparing the densities.

A submarine has tanks that it can fill with sea water or empty by pushing the water out, as shown in the figure.

1. a) When the submarine wants to submerge, it fills the tanks with water. Explain what happens to its average density and why this causes it to sink.
2. b) What does it do with the tanks when it wants to return to the surface? Justify your answer using average density.
3. c) Fish do something similar with the swim bladder, an organ they inflate or deflate to rise or descend in the water. Explain the similarity with the submarine.

At the supermarket, a $5\ \text{L}$ jug of water weighs $5\ \text{kg}$ (not counting the container), but a $5\ \text{L}$ jug of oil clearly weighs less.

1. a) Remember that $1\ \text{L} = 1000\ \text{cm}^3$. Calculate the mass of $5\ \text{L}$ of water ($1\ \text{g/cm}^3$) and check that it is $5\ \text{kg}$.
2. b) Calculate the mass of $5\ \text{L}$ of oil ($0,9\ \text{g/cm}^3$) and express it in $\text{kg}$.
3. c) Explain the result using density and comment on why the phrase "a litre weighs a kilo" is only true for water.

Unit challenge. What metal are your coins really made of? Become a scientific expert by investigating whether the density of $0,05\text{ €}$ coins matches the value expected from its official composition.

1. Official search: Check the Banco de España website for the official mass and composition of the $0,05\text{ €}$ coin.

2. Data collection with uncertainty: Use a kitchen scale (sensitivity $\pm 1\text{ g}$) and a graduated cylinder with water (sensitivity $\pm 0,5\text{ mL}$). Measure the mass and volume by immersion of groups of $10, 20, 30$, and $40$ identical coins, always recording the uncertainty of your instruments.

3. Processing and analysis: Prepare a data table ($m$ versus $V$) and draw the corresponding graph. Determine the slope to obtain the experimental density in $\text{g/cm}^3$ and compare it with the tabulated density of the material given in the official composition.

4. Scientific report: Write a reasoned conclusion justifying why mass and volume are general properties, while density is a characteristic property for identifying a material.

Assessment criteria:

• Rigorous recording of measurements with units and instrumental uncertainties.

• Correct graphical representation, determination of density from the slope and reasoned assessment of its uncertainty.

• Clear justification of the difference between general and characteristic properties based on the data.