Determining Density of Regular and Irregular Solids

Good to see you again. You can already calculate density from mass and volume; now the focus is the practical-investigation question: choosing a method, recording data, drawing the correct graph, and obtaining density from its gradient.

For a full-mark answer, keep one idea at the centre:

$$\rho =\frac{m}{V}$$

If you plot mass against volume, the graph’s gradient is also:

$$\frac{\Delta m}{\Delta V}$$

So the gradient is the density.

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1. What a density investigation is testing

A density investigation tests the relationship between the mass and volume of samples of the same material.

For a uniform material at constant temperature, density stays constant. Therefore:

• a larger volume contains more material;
• more material has a greater mass;
• mass is directly proportional to volume.

The investigation is designed with:

Type of variable	In a mass-volume investigation
Independent variable	Volume, $V$
Dependent variable	Mass, $m$
Key control variables	Same material; constant temperature; same measuring method and equipment

For solid samples, you would use several objects made from the same substance but with different volumes. If you mixed materials, their densities could differ, so one straight-line relationship would no longer be expected.

The prediction is:

As the volume of a sample increases, its mass increases in direct proportion, provided it is the same material.

The reason is not that “large objects are always dense.” It is that a larger sample contains more particles, so it has more mass. Its density remains unchanged.

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2. Choosing the volume method for a solid

Mass is straightforward for any solid: measure it with a top-pan balance, recording the value in grams or kilograms.

The method for volume depends on the object’s shape.

Regular-shaped solids

A regular solid has dimensions that can be measured and inserted into a geometric formula. Examples include a cuboid, cube, or cylinder.

For a rectangular block:

$$V=lwh$$

Measure length, width, and height with a ruler or vernier calipers, then multiply them.

For example, a metal block has:

$$l=5.0\mathrm{cm}$$ $$w=2.0\mathrm{cm}$$ $$h=1.5\mathrm{cm}$$

Its volume is:

$$V=5.0\times 2.0\times 1.5$$ $$V=15\mathrm{c}{\mathrm{m}}^3$$

Then measure its mass on the balance and use the graph data or $\rho =m/V$ to find density.

Irregular-shaped solids

An irregular solid, such as a stone, cannot have its volume reliably found using length × width × height. Its shape does not fit a simple formula.

Use water displacement instead. When a solid is fully submerged, it pushes aside a volume of water equal to its own volume.

For a small irregular object, use a measuring cylinder:

1. Add water and record the initial volume.
2. Carefully lower the solid into the water.
3. Record the final volume.
4. Subtract initial volume from final volume.

$$V_{\text{object}}=V_{\text{final}}-V_{\text{initial}}$$

If water rises from $50\mathrm{c}{\mathrm{m}}^3$ to $75\mathrm{c}{\mathrm{m}}^3$:

$$V_{\text{object}}=75-50$$ $$V_{\text{object}}=25\mathrm{c}{\mathrm{m}}^3$$

For a large irregular object that will not fit into a measuring cylinder, use a displacement can. Fill it until water stops leaving the spout, place a measuring cylinder under the spout, then fully submerge the object. The collected water volume equals the object’s volume.

Method-choice checklist

Object or substance	Measure mass with	Find volume using
Regular cuboid or cube	Top-pan balance	Ruler or calipers; calculate from dimensions
Small irregular solid	Top-pan balance	Measuring-cylinder displacement
Large irregular solid	Top-pan balance	Displacement can
Liquid	Top-pan balance and container	Measuring cylinder

A displacement method only works properly if the solid is fully submerged and does not absorb, dissolve in, or react with the liquid.

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3. Planning a strong method and recording results

For regular solids, a clear method would be:

1. Obtain several blocks of the same material with different sizes.
2. Measure the mass of each block using a top-pan balance.
3. Measure each block’s dimensions.
4. Calculate each volume.
5. Record all results in a table.
6. Plot mass against volume.
7. Draw a best-fit line and calculate its gradient.

A suitable results table might look like this:

Block	Mass / $\mathrm{g}$	Length / $\mathrm{cm}$	Width / $\mathrm{cm}$	Height / $\mathrm{cm}$	Volume / $\mathrm{c}{\mathrm{m}}^3$
1	27	2.0	2.0	2.5	10
2	54	2.0	3.0	3.3	20
3	80	3.0	2.0	5.0	30
4	109	4.0	2.0	5.0	40
5	135	5.0	3.0	3.3	50

Notice that every heading includes both the quantity and its unit. That is an easy mark in a practical question.

For an irregular solid, the table would instead include initial and final water readings:

Sample	Mass / $\mathrm{g}$	Initial volume / $\mathrm{c}{\mathrm{m}}^3$	Final volume / $\mathrm{c}{\mathrm{m}}^3$	Sample volume / $\mathrm{c}{\mathrm{m}}^3$
1	35	40	45	5
2	70	40	50	10

For each sample:

$$V_{\text{sample}}=V_{\text{final}}-V_{\text{initial}}$$

Improving the data

The most useful improvements are specific to the measurement being made:

• Repeat ruler or caliper measurements and calculate a mean, especially for small dimensions.
• Use a larger sample or larger volume where possible. A fixed reading uncertainty then has a smaller percentage effect.
• Zero the balance before use.
• Remove air bubbles from an irregular object during displacement.
• Ensure the object is fully submerged without splashing water out.
• Repeat suspicious readings rather than simply deleting them.

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4. Measuring a liquid correctly

A liquid’s volume is measured with a measuring cylinder. Its mass must be found without including the container’s mass.

There are two equivalent approaches:

1. Place an empty measuring cylinder on the balance and tare it to zero; then add the liquid.
2. Measure the empty cylinder’s mass, measure the cylinder plus liquid, then subtract.

$$m_{\text{liquid}}=m_{\text{cylinder + liquid}}-m_{\text{empty cylinder}}$$

To obtain an accurate volume, place the measuring cylinder on a flat bench and read the bottom of the meniscus at eye level.

Do not add liquid while the measuring cylinder is sitting on an electronic balance. Remove it, add the liquid, wipe any drips if needed, then return it to the balance.

For a fair liquid investigation, keep the liquid and its temperature constant. Temperature can change density; you will examine why in the next lesson.

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5. The graph that gives density

The required graph is a graph of mass against volume. In science, “$y$ against $x$” means:

• vertical $y$-axis: mass;
• horizontal $x$-axis: volume.

So label your axes:

$$\text{Mass / g}$$

on the vertical axis, and:

$${\text{Volume / cm}}^3$$

on the horizontal axis.

A high-quality graph should have:

• a sensible, even scale that uses most of the graph paper;
• clearly labelled axes with units;
• neat plotted crosses or points;
• a line of best fit, not dot-to-dot joins;
• a title if your test or teacher requires one.

For samples of one uniform material, the expected pattern is a straight line through the origin. At zero volume, there should be zero mass of that material.

If your points scatter around a line, that is normal measurement variation. Draw the line so that it passes as centrally as possible through the overall trend; do not try to force it through every point.

A point very far from the rest may be anomalous. Check whether there was a recording mistake, a wrongly calculated volume, spilled water, an air bubble, or a balance-reading error. Repeat the measurement if possible.

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6. Calculating density from the best-fit gradient

The gradient of any graph is:

$$\text{gradient}=\frac{\Delta y}{\Delta x}$$

On this graph:

$$\text{gradient}=\frac{\Delta m}{\Delta V}$$

But density is:

$$\rho =\frac{m}{V}$$

Therefore:

$$\text{gradient}=\rho$$

The graph gradient has the correct density unit:

$$\frac{\mathrm{g}}{\mathrm{c}{\mathrm{m}}^3}=g/c{\mathrm{m}}^3$$

Worked graph calculation

Suppose your line of best fit passes through these two convenient points:

$$(0\mathrm{c}{\mathrm{m}}^3,0\mathrm{g})$$

and

$$(50\mathrm{c}{\mathrm{m}}^3,135\mathrm{g})$$

Use the points on the line, even if they are not original data points:

$$\text{gradient}=\frac{135-0}{50-0}$$ $$\text{gradient}=2.7g/c{\mathrm{m}}^3$$

Therefore, the density of the material is:

$$\rho =2.7g/c{\mathrm{m}}^3$$

Choose points that are:

• on the best-fit line;
• far apart;
• easy to read from the grid.

Using points far apart reduces the effect of small reading errors. Do not calculate the gradient by choosing two random experimental crosses if the question asks for the best-fit gradient.

How to write the conclusion

A complete conclusion for this investigation is:

As volume increased, mass increased in direct proportion. The mass-volume graph was a straight line through the origin. The gradient of the best-fit line was $2.7g/c{\mathrm{m}}^3$, so the density of the material was $2.7g/c{\mathrm{m}}^3$.

If the line has a noticeable non-zero intercept when theory says it should pass through the origin, suspect a systematic error. For example, in a liquid experiment, the container’s mass may not have been subtracted or the balance may not have been tared.

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7. A test-day answer structure

When asked to plan or analyse a density practical, use this order:

1. State the equipment: balance plus ruler/calipers or displacement apparatus.
2. Explain how mass is measured.
3. Explain how volume is measured, matching the method to the shape.
4. State variables: volume independent, mass dependent, material and temperature controlled.
5. Describe the graph: mass on $y$, volume on $x$, both with units, best-fit line.
6. Find density from gradient:

$$\rho =\frac{\Delta m}{\Delta V}$$

7. Give a conclusion about direct proportionality, including the density and unit.
8. Suggest realistic improvements linked to likely errors.

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You now have the practical-density toolkit: calculate volume from dimensions for a regular object, use water displacement for an irregular object, measure mass with a balance, and plot mass against volume. The decisive graph fact is:

$$\text{gradient of mass-volume graph}=\text{density}$$

Next, you will connect density to temperature, changes of state, and whether objects float or sink.