Measurements in physics: length, volume, mass, time and density

A tailor measures cloth, a boda boda rider checks fuel in litres, a milk collector tests whether milk has been mixed with water. All of them are doing physics: measuring.

By the end of these notes you should be able to:

• estimate and measure length, area, volume, mass and time using the right instruments and units;
• read a metre rule, vernier calipers and a micrometer screw gauge accurately;
• reduce errors by repeating measurements and averaging;
• use scientific notation and significant figures;
• find density and use it to explain floating, sinking and purity.

Physical quantities and SI units

A physical quantity is anything that can be measured, such as length or mass. Every measurement has a number and a unit: "5" means nothing, but "5 m" does.

Scientists around the world use the International System of Units (SI). The basic quantities and their units are:

Quantities such as area, volume and density are derived quantities: they are worked out from basic ones.

Prefixes

Measuring length

Choosing the right instrument

Using a metre rule

• Place the rule flat against the object.
• Keep your eye directly above the mark you are reading. Looking from the side causes parallax error.
• If the end of the rule is worn, start measuring from the 10 cm mark and subtract 10 cm.

Reading vernier calipers

Vernier calipers have a main scale in centimetres and millimetres and a sliding vernier scale with 10 divisions.

1. Read the main scale just before the zero of the vernier scale. This gives cm and mm.
2. Find the vernier mark that lines up exactly with a main-scale mark. This gives the hundredths of a cm.
3. Add the two readings.

Reading a micrometer screw gauge

The sleeve shows millimetres and half-millimetres. The thimble has 50 divisions, each worth 0.01 mm.

Measuring area

For regular shapes, measure the sides and use a formula: rectangle = length × width; triangle = ½ × base × height; circle = πr². For irregular shapes, such as a leaf, trace the outline on graph paper and count the squares: count every full square and every square more than half covered.

Measuring volume

• Regular solids: measure and calculate. A cuboid = length × width × height.
• Liquids: use a measuring cylinder, pipette or burette. Read the bottom of the curved surface (meniscus) with your eye level with it.
• Irregular solids (a stone, a key): use the displacement method. Put water in a measuring cylinder and note the level. Lower the stone in gently on a thread. The rise in the water level equals the volume of the stone. For large objects, use a eureka (overflow) can and measure the water that overflows.

Units: 1 cm³ = 1 ml; 1000 cm³ = 1 litre; 1 m³ = 1 000 000 cm³ = 1000 litres.

Measuring mass and time

Mass is the amount of matter in a body, measured in kilograms with a beam balance or electronic balance. It is not the same as weight, which is the pull of gravity on that mass (you will study this later).

Time is measured with a stopwatch. Human reaction time causes errors of about 0.2 s, so time many events and divide.

Errors and accuracy

No measurement is perfect. Common sources of error:

• Parallax error: reading a scale from an angle.
• Zero error: an instrument that does not read zero when it should.
• Reaction time: starting or stopping a stopwatch late.
• Instrument limits: a metre rule cannot measure a wire's thickness accurately.

To improve accuracy: choose the right instrument, read scales at eye level, check for zero error, and repeat measurements and take the average. An average reduces the effect of random errors.

The scientific method

Physics answers questions by investigation:

1. Ask a question. Does a longer pendulum swing more slowly?
2. Make a prediction (hypothesis). A longer pendulum takes more time per swing.
3. Plan a fair test: change only one variable (length), measure one variable (time), and keep everything else the same (mass of bob, size of swing).
4. Carry out the experiment and record results in a table.
5. Analyse: draw a graph and look for a trend.
6. Conclude and justify your answer using the results. Suggest improvements.

Scientific notation and significant figures

Scientific notation (standard form) writes very large or very small numbers as A × 10ⁿ, where A is between 1 and 10.

• Distance from the Earth to the sun: 150 000 000 km = 1.5 × 10⁸ km.
• Thickness of a hair: 0.00008 m = 8 × 10⁻⁵ m.

Significant figures show how precise a measurement is. Count from the first non-zero digit. Zeros between digits and zeros at the end after a decimal point count.

Density

Density is the mass per unit volume of a substance.

Density = mass ÷ volume, written ρ = m ÷ V.

SI unit: kg/m³. We also use g/cm³. To change g/cm³ into kg/m³, multiply by 1000. Water has a density of 1 g/cm³ = 1000 kg/m³.

Floating and sinking

An object floats in a liquid if its density is less than the liquid's density, and sinks if it is greater. Ice floats on water; a stone sinks. Oil floats on water, which is why oil spills spread across the surface of lakes.

A steel ship floats even though steel is denser than water, because the ship is hollow: its average density, including the air inside, is less than that of water.

Density and purity

Every pure substance has its own density, so density can test purity:

• Milk: milk collectors use a lactometer, which measures density. Adding water lowers the density of milk below its normal value of about 1.03 g/cm³, so watered milk is detected.
• Gold: if a "gold" ring has a density much lower than 19.3 g/cm³, it is mixed with cheaper metals.

Summary

• Measurements need a number and a unit; use SI units and convert before calculating.
• Choose instruments by the size and precision needed: tape measure, metre rule, vernier calipers (0.01 cm), micrometer (0.01 mm).
• Avoid parallax and zero errors; repeat and average to improve accuracy.
• Use scientific notation for very large or small numbers, and sensible significant figures.
• Density = mass ÷ volume. Less dense objects float; density also tests purity.

Practice questions

1. Name the most suitable instrument for measuring (a) the diameter of a copper wire (b) the length of a classroom (c) the internal diameter of a test tube.
2. Convert: (a) 2.5 km to m (b) 450 mm to m (c) 3.2 g/cm³ to kg/m³.
3. Write in scientific notation: (a) 0.00032 (b) 6 400 000.
4. A block of wood measures 10 cm × 5 cm × 4 cm and has a mass of 140 g. Find its density in g/cm³ and in kg/m³. Will it float in water? Explain.
5. Describe how you would find the volume of an irregular stone too large for a measuring cylinder.
6. State two ways of reducing errors when timing a pendulum.
7. Activity of integration. A dairy cooperative in Mbarara suspects that some farmers add water to their milk. Explain to the cooperative how density can be used to detect this, and describe a simple test they could carry out each morning.

Answers: 2) (a) 2500 m (b) 0.45 m (c) 3200 kg/m³ 3) (a) 3.2 × 10⁻⁴ (b) 6.4 × 10⁶ 4) 0.7 g/cm³ = 700 kg/m³; it floats because it is less dense than water (1 g/cm³).