Measuring Density

Density is not usually measured directly.

Instead, we measure two quantities:

• mass
• volume

We then use these measurements to calculate:

density.

The quality of our density result depends on how accurately we measure both mass and volume.

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The Density Equation

Density is defined as:

density = mass ÷ volume

or:

ρ = m / V

where:

ρ = density

m = mass

V = volume

Common units include:

g/cm³

g/mL

and:

kg/m³

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A Basic Density Investigation

To determine the density of an unknown object:

1. Measure its mass.

2. Determine its volume.

3. Calculate its density.

4. Record the answer with appropriate units and precision.

The method used to determine volume depends on whether the object is:

regular or irregular.

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Measuring Mass

Mass is measured using a:

balance.

Common laboratory balances include:

• digital balances
• electronic top-pan balances
• triple-beam balances

For most school laboratory investigations, an electronic balance provides a quick and precise measurement.

Using a Digital Balance Correctly

A good procedure is:

1. Place the balance on a stable, level surface.
2. Switch it on.
3. Check that the display reads 0.
4. If necessary, press tare or zero.
5. Place the object gently on the balance.
6. Wait for the reading to stabilize.
7. Record the measurement and its unit.

For example:

Mass = 56.4 g

Do not record simply:

56.4

A measurement without a unit is:

incomplete.

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What Does "Tare" Mean?

The tare function resets a balance to zero while something is sitting on it.

This is useful when measuring a substance inside a:

container.

Suppose an empty beaker is placed on the balance.

The balance reads:

82.6 g.

Press tare.

The display returns to:

0.0 g.

Now add the liquid.

The balance displays the mass of the:

liquid alone.

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Measuring the Mass of a Liquid

Liquids cannot normally be placed directly onto a balance.

Instead:

1. Place an empty container on the balance.
2. Record its mass.
3. Add the liquid.
4. Measure the combined mass.
5. Subtract the container mass.

For example:

Mass of empty container = 42.3 g

Mass of container + liquid = 87.8 g

Mass of liquid:

87.8 − 42.3 = 45.5 g

Alternatively, use the tare function before adding the liquid.

Choosing the Correct Balance

Not every balance has the same:

precision.

One balance might measure to:

1 g

another to:

0.1 g

and another to:

0.01 g.

For a small object, a balance capable of measuring smaller mass differences usually provides a more useful result.

However, the object must also be within the balance's:

maximum capacity.

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Measuring Volume

Volume is the amount of space occupied by an object or substance.

The method used depends on the:

shape and state of the material.

We can divide measurements into:

regular solids

irregular solids

and:

liquids.

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Measuring a Regular Solid

A regular solid has a shape whose volume can be calculated using a mathematical formula.

Examples include:

• rectangular blocks
• cubes
• cylinders
• spheres

For these objects, measure the necessary dimensions using a ruler, meter stick, or:

caliper.

Rectangular Objects

For a rectangular block:

Volume = length × width × height

or:

V = l × w × h

Suppose:

length = 5.0 cm

width = 4.0 cm

height = 2.0 cm

Then:

V = 5.0 × 4.0 × 2.0

V = 40 cm³

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Worked Example 1 — Regular Block

A block has dimensions:

6.0 cm × 3.0 cm × 2.0 cm

Its mass is:

97.2 g.

First calculate volume:

V = 6.0 × 3.0 × 2.0

V = 36 cm³

Then calculate density:

ρ = 97.2 ÷ 36

ρ = 2.7 g/cm³

The measured density is:

2.7 g/cm³.

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Measuring Cylinders

For a cylindrical object:

V = πr²h

where:

r = radius

and:

h = height.

The diameter can be measured and divided by 2 to determine the:

radius.

A caliper can often measure diameter more precisely than an ordinary ruler.

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Worked Example 2 — Cylinder

A cylinder has:

radius = 2.0 cm

height = 5.0 cm

Mass = 188 g

Volume:

V = π(2.0)²(5.0)

V ≈ 62.8 cm³

Density:

ρ = 188 ÷ 62.8

ρ ≈ 2.99 g/cm³

With appropriate significant figures:

ρ ≈ 3.0 g/cm³

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Measuring an Irregular Solid

Some objects do not have a simple geometric shape.

Examples include:

• rocks
• metal bolts
• keys
• irregular mineral samples

Their volume can often be measured using:

water displacement.

Water Displacement

The method is based on a simple idea:

A completely submerged object displaces a volume of water equal to the object's volume.

For example:

Initial water level = 45 mL

Final water level = 68 mL

Object volume:

68 − 45 = 23 mL

Because:

1 mL = 1 cm³

the object's volume is:

23 cm³.

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Water Displacement Procedure

Step 1

Add enough water to a graduated cylinder to completely cover the object.

Step 2

Record the initial volume.

Step 3

Carefully lower the object into the water.

Step 4

Make sure it is completely submerged.

Step 5

Record the final volume.

Step 6

Calculate:

Object volume = final volume − initial volume

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Worked Example 3 — Irregular Rock

A rock has a mass of:

74.6 g.

Initial water level:

50.0 mL

Final water level:

78.0 mL

Volume:

78.0 − 50.0 = 28.0 mL

Therefore:

V = 28.0 cm³

Density:

ρ = 74.6 ÷ 28.0

ρ = 2.66 g/cm³

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Reading a Graduated Cylinder

Correctly reading a graduated cylinder is important.

For many liquids, including water, the surface curves slightly.

This curved surface is called the:

meniscus.

For water, the volume is normally read from the:

bottom of the meniscus.

Avoiding Parallax Error

Your eye should be:

level with the meniscus.

Looking from above or below can make the liquid appear to line up with the wrong scale marking.

This creates:

parallax error.

Correct:

eye level with the meniscus

Incorrect:

looking down or up at the scale.

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Choosing an Appropriate Measuring Cylinder

Suppose you need to measure approximately:

25 mL.

A 50 mL graduated cylinder may give a more precise measurement than a 1000 mL cylinder.

Why?

The smaller cylinder usually has:

finer scale divisions.

Choose measuring equipment that provides sufficient precision for the quantity being measured.

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Measuring Liquid Volume

The volume of a liquid can be measured directly using equipment such as:

• graduated cylinders
• pipettes
• burettes
• volumetric flasks

For a basic density investigation, a:

graduated cylinder

is usually suitable.

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Measuring Liquid Density

To measure the density of a liquid, you need:

mass of the liquid

and:

volume of the liquid.

A simple method is:

1. Measure the mass of an empty graduated cylinder.
2. Add a measured volume of liquid.
3. Measure the mass of the cylinder and liquid.
4. Subtract the empty cylinder mass.
5. Calculate density.

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Worked Example 4 — Liquid Density

Mass of empty cylinder:

36.2 g

Mass of cylinder + liquid:

76.7 g

Liquid volume:

50.0 mL

Mass of liquid:

76.7 − 36.2 = 40.5 g

Density:

ρ = 40.5 ÷ 50.0

ρ = 0.810 g/mL

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Recording Measurements

Good scientists do not simply write down numbers.

They record:

quantity + numerical value + unit

For example:

A well-designed data table makes measurements easier to:

interpret and check.

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Put Units in Table Headings

Instead of repeatedly writing:

48.6 g

51.2 g

47.9 g

a scientific table can use a heading such as:

Mass / g

Then the data column contains:

48.6

51.2

47.9

This makes the table clear and avoids unnecessary repetition.

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Significant Figures

Measurements are limited by the precision of the equipment used.

Significant figures communicate the precision of a measured or calculated quantity.

For example:

4 g

4.0 g

4.00 g

do not communicate the same measurement precision.

The extra zeros indicate that the quantity was measured to a finer:

precision.

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Identifying Significant Figures

Examples:

23.4 → 3 significant figures

5.62 → 3 significant figures

0.0045 → 2 significant figures

2.00 → 3 significant figures

1500 can be ambiguous unless additional notation is used.

Scientific notation can remove this ambiguity:

1.5 × 10³ → 2 significant figures

1.500 × 10³ → 4 significant figures

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Significant Figures in Density Calculations

Suppose:

Mass = 24.6 g

Volume = 9.2 cm³

Calculator result:

24.6 ÷ 9.2 = 2.673913...

It would be inappropriate to report:

2.673913 g/cm³

because the original measurements were not that precise.

The least precise measurement has:

2 significant figures.

Therefore report:

2.7 g/cm³.

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Why Calculator Displays Can Be Misleading

A calculator might display:

3.428571429

That does not mean your experiment measured density to ten significant figures.

The precision of the final answer depends on the precision of the:

measurements.

The calculator cannot make experimental data more precise.

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Precision and Accuracy

These terms are related but different.

Accuracy describes how close a measurement is to the accepted or true value.

Precision describes how closely repeated measurements agree with each other and, in another common sense, the resolution with which a measurement is recorded.

A set of measurements can be:

• accurate and precise
• accurate on average but not very precise
• precise but inaccurate
• neither accurate nor precise

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Repeated Measurements

One way to improve the reliability of an investigation is to take:

repeated measurements.

For example:

These measurements are very close together.

An average can be calculated:

Mean = (2.68 + 2.71 + 2.69) ÷ 3

Mean ≈ 2.69 g/cm³

Repeating measurements can help identify:

random variation and anomalous results.

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What Is Measurement Uncertainty?

No experimental measurement is perfectly exact.

Every measuring instrument has a limit to how precisely it can:

measure.

For example, a ruler marked every millimetre cannot reliably provide unlimited decimal places.

Similarly, a balance displaying to:

0.1 g

cannot justify reporting mass to:

0.0001 g.

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Sources of Error

A good density investigation should include an:

evaluation.

Evaluation means considering what could have affected the results and how the investigation could be:

improved.

Several common sources of error can affect density measurements.

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Error 1 — Balance Not Zeroed

If the balance does not read zero before measurement, every mass measurement may be:

shifted.

This could make the calculated density systematically too high or too low.

Improvement

Check the balance and:

tare or zero it before measuring.

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Error 2 — Incorrect Meniscus Reading

Reading the liquid level from the wrong position can produce an incorrect:

volume.

Improvement

Read the bottom of the water meniscus at:

eye level.

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Error 3 — Parallax

Looking at a scale from an angle can make the reading appear:

different.

Improvement

Position your eye directly level with the measurement mark.

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Error 4 — Air Bubbles

When an irregular object is submerged, air bubbles may become trapped on its:

surface.

The bubbles also displace water.

This can make the measured object volume appear:

too large.

If volume is too large, the calculated density becomes:

too low.

Improvement

Gently move or tap the object to release trapped bubbles before taking the final reading.

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Error 5 — Object Not Fully Submerged

If part of an irregular object remains above the water, it will not displace its full:

volume.

The measured volume will be too small.

The calculated density will therefore be:

too high.

Improvement

Ensure the object is completely submerged.

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Error 6 — Water Absorption

Some materials, such as certain rocks, woods, or porous materials, may absorb:

water.

This can affect both the object's mass and the apparent displacement measurement.

Improvement

Choose a suitable method for the material or minimize the time it remains submerged.

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Error 7 — Water Remaining on the Object

If an object is weighed after being removed from water, droplets may remain attached.

This makes its measured mass:

too high.

Improvement

Dry the object appropriately before measuring its mass.

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Error 8 — Poor Dimension Measurements

For a regular object, inaccurate measurements of length, width, or height affect the calculated:

volume.

Because several dimensions may be multiplied together, small measurement errors can influence the final result.

Improvement

Use appropriate equipment and measure carefully.

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Ruler or Caliper?

A ruler may be suitable for a large rectangular block.

For a small object, a:

vernier caliper

or digital caliper may provide greater precision.

The best instrument depends on:

• object size
• shape
• required precision
• available equipment

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Error 9 — Spilling Liquid

If liquid is spilled during an experiment, the measured mass and volume may no longer represent the same:

sample.

Improvement

Transfer liquids carefully and repeat the measurement if a spill occurs.

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Error 10 — Limited Instrument Resolution

Suppose a graduated cylinder has markings every:

10 mL.

Trying to measure a volume change of only:

2 mL

would produce a large relative uncertainty.

Improvement

Use a smaller cylinder with finer graduations or use a larger sample.

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Why Larger Samples Can Help

Suppose an instrument has an uncertainty of roughly the same absolute size in every measurement.

An error of 1 mL is very significant when measuring:

5 mL.

It is much less significant when measuring:

100 mL.

Using a suitably larger sample can therefore reduce the:

percentage uncertainty.

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Percentage Error

If an accepted value is known, experimental accuracy can be evaluated using percentage error:

Percentage error = |experimental − accepted| ÷ accepted × 100%

The vertical bars mean we use the:

absolute difference.

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Worked Example 5 — Percentage Error

Experimental density:

2.62 g/cm³

Accepted density:

2.70 g/cm³

Difference:

|2.62 − 2.70| = 0.08

Percentage error:

0.08 ÷ 2.70 × 100

≈ 2.96%

Rounded appropriately:

≈ 3.0%

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Systematic and Random Errors

Experimental errors can often be considered in two broad categories.

Random errors

These vary unpredictably between measurements.

Examples include:

• slightly different meniscus readings
• small changes in positioning
• minor fluctuations in balance readings

Repeating measurements and calculating a mean can reduce the influence of:

random error.

Systematic errors

These consistently shift measurements in one direction.

Examples include:

• a balance that is incorrectly calibrated
• a ruler with a damaged zero point
• consistently reading a scale incorrectly

Repeating measurements does not necessarily remove:

systematic error.

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Evaluating an Experiment Properly

A weak evaluation says:

"There may have been human error."

This is too vague.

A stronger evaluation identifies:

the specific error + its effect + an improvement.

For example:

Air bubbles may have remained attached to the rock. This would increase the measured displaced volume and make the calculated density too low. The object should be gently moved underwater to remove trapped bubbles before recording the final volume.

This demonstrates scientific:

reasoning.

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Error Direction Matters

Good evaluation should consider whether an error makes a result:

too high or too low.

Suppose:

density = mass ÷ volume.

If measured mass is too high:

calculated density is too high.

If measured mass is too low:

calculated density is too low.

If measured volume is too high:

calculated density is too low.

If measured volume is too low:

calculated density is too high.

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Error Analysis Table

Being able to predict these effects is an important experimental:

skill.

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Designing a Good Density Investigation

A strong procedure might include:

1. Select an appropriate object.
2. Choose suitable measuring equipment.
3. Zero the balance.
4. Measure mass.
5. Determine volume using the appropriate method.
6. Repeat measurements where practical.
7. Record data systematically.
8. Calculate density.
9. Report appropriate units and significant figures.
10. Compare with an accepted value if available.
11. Identify uncertainties and sources of error.
12. Suggest realistic improvements.

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Example Data Table — Regular Solids

A good table clearly identifies:

variables and units.

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Example Data Table — Irregular Solids

Notice that two objects can have different masses and volumes but still have the same:

density.

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Worked Example 6 — Complete Investigation

A student investigates an irregular metal sample.

Mass:

156.2 g

Initial water volume:

40.0 mL

Final water volume:

60.0 mL

Step 1 — Determine volume

V = 60.0 − 40.0

V = 20.0 cm³

Step 2 — Calculate density

ρ = 156.2 ÷ 20.0

Calculator:

7.81 g/cm³

Step 3 — Report result

The sample has a density of:

7.81 g/cm³

The result is close to the density expected for some types of:

iron or steel.

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Worked Example 7 — Spotting a Problem

A student records:

Mass = 65.2 g

Volume = 24 cm³

Density = 2.7166666667 g/cm³

What is wrong?

The calculation itself is reasonable, but the answer contains:

far too many significant figures.

Because the volume is recorded to only two significant figures, an appropriate result would be:

2.7 g/cm³.

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Worked Example 8 — Evaluating Air Bubbles

A student measures a rock using water displacement.

Several air bubbles remain attached to the rock.

What happens?

The bubbles displace additional:

water.

Measured volume becomes:

too large.

Since:

density = mass ÷ volume

the calculated density becomes:

too low.

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Worked Example 9 — Incomplete Submersion

A rock is only partly submerged when the final water level is recorded.

The measured displaced volume will be:

too small.

Therefore the calculated density will be:

too high.

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Worked Example 10 — Selecting Equipment

A student needs to measure the volume of a small metal cube approximately 1 cm wide.

Would a ruler marked only in centimetres be ideal?

No.

The object is small, so the relative uncertainty would be large.

A:

caliper

or ruler with millimetre divisions would provide a more precise measurement.

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Common Mistake: Forgetting to Zero the Balance

Always check:

Does the balance read zero before measurement?

If not, zero or tare it.

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Common Mistake: Forgetting the Container Mass

When measuring liquid mass:

mass of container + liquid ≠ mass of liquid.

Either subtract the container mass or:

tare the balance first.

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Common Mistake: Reading the Top of the Meniscus

For water and many common liquids in glassware, read the:

bottom of the meniscus.

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Common Mistake: Looking Down at the Cylinder

Read the liquid level at:

eye level.

This reduces parallax error.

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Common Mistake: Using Final Volume as Object Volume

Suppose:

Initial = 40 mL

Final = 65 mL

The object's volume is not:

65 cm³.

It is:

65 − 40 = 25 cm³.

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Common Mistake: Reporting Too Many Decimal Places

Your calculator may display many digits.

Your experimental measurements do not justify all of them.

Report the final value using appropriate:

significant figures.

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Common Mistake: Saying "Human Error"

An evaluation should be specific.

Instead of:

"Human error affected the results."

identify:

• what happened
• which measurement was affected
• whether the density became too high or too low
• how the method could be improved

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Practical Investigation — Density Detective

This topic is especially suitable for a laboratory investigation.

Equipment

• digital balance
• graduated cylinder
• water
• ruler
• caliper if available
• several regular objects
• several irregular objects
• paper towels

Challenge

Determine the density of each unknown object and use a reference table to suggest what material it might be made from.

For each sample, students should record:

mass

volume

density

measurement method

possible material

and:

sources of uncertainty.

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Extension — Which Method Is Better?

For a regular metal cylinder, you could determine volume in two ways:

Method A: Measure its dimensions and calculate volume.

Method B: Use water displacement.

Perform both methods and compare the calculated densities.

Ask:

• Are the results identical?
• Which method is more precise?
• Which measurements have the greatest uncertainty?
• Which method is easier?
• What experimental errors affect each method?

This turns a simple density calculation into an investigation of:

experimental quality.

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Check Your Understanding

1. What two measurements are needed to calculate density?
2. What instrument is used to measure mass?
3. What does the tare function do?
4. Why should a balance be zeroed before use?
5. How can you measure the mass of a liquid?
6. Why must the container mass be considered?
7. How can you determine the volume of a rectangular block?
8. Write the volume equation for a rectangular prism.
9. How can the volume of a cylinder be calculated?
10. Why might a caliper be better than a ruler for a small object?
11. How can you determine the volume of an irregular rock?
12. Explain the principle of water displacement.
13. Water rises from 36.0 mL to 59.0 mL. What is the object's volume?
14. Why is 1 mL equivalent to 1 cm³?
15. What is a meniscus?
16. Where should the water meniscus normally be read?
17. Why should your eye be level with the liquid surface?
18. What is parallax error?
19. Why should you choose an appropriately sized graduated cylinder?
20. Describe how to measure the density of a liquid.
21. A block has a mass of 81 g and volume of 30 cm³. Calculate its density.
22. A rock has mass 54.6 g. Water rises from 25.0 mL to 45.0 mL. Calculate its density.
23. An empty cylinder has mass 40.2 g. With 50.0 mL of liquid it has mass 82.7 g. Calculate the liquid's density.
24. Why must units always be recorded?
25. What are significant figures?
26. How many significant figures are in 4.52?
27. How many significant figures are in 0.0062?
28. How many significant figures are in 3.00?
29. Why should a calculator result not automatically be copied in full?
30. Explain the difference between accuracy and precision.
31. Why are repeated measurements useful?
32. What is measurement uncertainty?
33. What happens if a balance is not correctly zeroed?
34. How can air bubbles affect a water-displacement measurement?
35. How would air bubbles affect calculated density?
36. What happens if an object is not completely submerged?
37. How would incomplete submersion affect calculated density?
38. Why can water absorption cause problems?
39. How can water droplets affect a mass measurement?
40. Why can poor dimension measurements strongly affect calculated volume?
41. What is a random error?
42. What is a systematic error?
43. Can repeated measurements eliminate a systematic error? Explain.
44. Why is "human error" a weak evaluation?
45. Give a specific error and explain its effect on calculated density.
46. How could you improve a density experiment involving a very small object?
47. What is percentage error?
48. Why might a larger sample reduce percentage uncertainty?
49. Design a method to determine the density of an unknown irregular metal object.
50. Explain how you would evaluate the quality of your final density result.

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Key Terms

Balance: Instrument used to measure mass.

Tare: Reset a balance to zero, often while a container is on it.

Volume: Amount of space occupied by an object or substance.

Density: Mass per unit volume.

Water displacement: Method for determining the volume of an irregular object using the change in liquid level.

Meniscus: Curved surface of a liquid in a container.

Parallax error: Measurement error caused by viewing a scale from an incorrect angle.

Significant figures: Digits used to communicate the precision of a measured or calculated value.

Accuracy: Closeness of a result to an accepted or true value.

Precision: Closeness of repeated measurements to one another, or the level of detail/resolution in a measurement.

Random error: Unpredictable variation between repeated measurements.

Systematic error: Consistent error that shifts measurements in the same direction.

Uncertainty: Range associated with the limitations of a measurement.

Percentage error: Difference between experimental and accepted values expressed as a percentage of the accepted value.

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Key Takeaways

• Density is determined by measuring mass and volume.
• Mass should be measured using an appropriate balance.
• Always check that the balance is zeroed or tared.
• Regular solids can have their volume calculated from measured dimensions.
• Irregular solids can often be measured using water displacement.
• Object volume = final water volume − initial water volume.
• Liquid volume should be measured using appropriately sized equipment.
• Water should normally be read at the bottom of the meniscus at eye level.
• Always record measurements with the correct units.
• The number of digits reported should reflect the precision of the measuring equipment.
• Calculators do not increase the precision of experimental measurements.
• Repeated measurements help identify random variation and improve reliability.
• Common density errors include incorrect balance zeroing, parallax, trapped air bubbles, incomplete submersion, poor dimension measurements, and inappropriate measuring equipment.
• A strong evaluation identifies the specific error, its effect on the result, and a realistic improvement.
• Understanding measurement quality is just as important as calculating the correct density.