Units, Measurement and Scientific Instruments

The International System of Units, abbreviated SI, provides shared measurement standards. Common units make scientific results and trade measurements easier to compare.

Different regions have used different measurement systems. Standard units reduce confusion, but instruments still need suitable calibration and careful use.

2 m and 200 cm describe the same length. Changing the unit changes the numerical value, not the object.

SI has seven base quantities and seven corresponding base units. Learn the quantity, unit name and symbol together.

The mole describes amount of substance, not mass. The candela describes luminous intensity, not total electrical power.

The kelvin uses K, without a degree sign. Write 300 K, not 300 °K. The degree Celsius uses °C.

Modern SI definitions use fixed numerical values of physical constants. A shared definition and an instrument's actual reading are different things.

Derived units combine base units through physical relationships. Speed is distance divided by time, so its unit is m/s.

The newton honours Isaac Newton. The henry honours Joseph Henry. Unit names are normally lowercase, even when their symbols use capitals.

H means henry; Hz means hertz. Inductance and frequency are different quantities. Halogen is a chemical-family term, not the unit represented by H.

Write 5 kg, not 5 kgs. Leave a space between the number and unit symbol.

Prefixes represent powers of ten. They change a unit's size, not the physical quantity being measured.

Square or cube the conversion factor

One centimetre equals 10⁻² m. Therefore, 1 cm² = 10⁻⁴ m² and 1 cm³ = 10⁻⁶ m³.

For example, 100 cm² = 100 × 10⁻⁴ m² = 0.01 m². Do not convert area using only the length factor.

Keep meaningful digits visible

Scientific notation uses a number multiplied by a power of ten. Write 0.00450 m as 4.50 × 10⁻³ m. Both forms show three significant figures.

Leading zeros locate the decimal point; they are not significant figures. The final zero in 4.50 records the reported precision.

For multiplication or division, a school-level rule uses the fewest significant figures among measured inputs. Exact counts and exact conversion factors do not limit the result. Keep extra digits during calculation and round the final answer.

Choose an instrument for the quantity and range you need. An instrument's name alone does not guarantee a reliable reading.

A seismograph produces a ground-motion record. It does not simply display one earthquake-magnitude number.

A voltmeter connects in parallel across the component being measured. An ammeter connects in series with the current path. These arrangements measure different electrical quantities.

Accuracy concerns closeness to a reference value. Precision concerns agreement among repeated measurements under stated conditions. Closely grouped readings can still be wrong together.

Suppose a reference length is 10.00 cm. An instrument repeatedly gives 9.80 cm, 9.81 cm and 9.80 cm. These readings are close together, but consistently below the reference.

Repeated measurements help reveal random variation. Averaging does not automatically remove a consistent instrument offset.

Check resolution and zero error

Resolution is the smallest change an instrument can distinguish. For a school scale, its least count describes the smallest directly readable increment. Calipers and micrometers often resolve smaller length changes than an ordinary ruler.

Finer divisions do not automatically guarantee better accuracy. Check calibration, zero error, alignment and the measuring technique.

For a screw gauge, least count = pitch ÷ circular-scale divisions. Pitch is the distance advanced in one full turn. If pitch is 0.5 mm with 50 divisions, the least count is 0.01 mm.

If an instrument reads +0.02 mm when it should read zero, subtract 0.02 mm from its indicated result. That corrects this stated zero offset; it does not eliminate every possible error.

Measurement uncertainty describes the doubt associated with a result. Reporting many calculator digits cannot remove uncertainty in the original measurement. Uncertainty can carry through later calculations.

Density is mass divided by volume. Its SI unit is kg/m³.

Relative density is a density ratio, using a stated reference. For common solid and liquid questions, the reference is water. Dividing equal units makes the ratio unitless.

Use density = relative density × reference density.

Suppose gold's relative density is 19.3 and the problem uses water's density as 1000 kg/m³.

19.3 × 1000 = 19300 kg/m³ = 1.93 × 10⁴ kg/m³.

The value 19.3 alone has no density unit. Water's 1000 kg/m³ is a useful rounded reference, not an exact density at every temperature.

The current doubles only under the stated fixed conditions. For a uniform wire, R = ρₑL/A. Here, ρₑ is resistivity, L is length and A is cross-sectional area.

Keep the material, temperature and cross-sectional area unchanged. Halving L then halves R.

At the same voltage across an ohmic wire, I = V/R. Halving the resistance doubles the current. Assume other circuit resistances and meter effects are negligible.

For example, V = 6 V and R = 3 Ω give I = 2 A. With the wire's resistance reduced to 1.5 Ω, the current becomes 4 A.

Stretching a wire is a different change because its cross-sectional area may also change.

Equal numerical readings do not make two temperature scales identical. Let C, F and K denote the numerical readings in °C, °F and kelvin.

Use K = C + 273.15 and F = 1.8C + 32.

If K = F, then C + 273.15 = 1.8C + 32.

0.8C = 241.15, so C = 301.4375.

The shared kelvin/Fahrenheit number is 574.5875. It is not the Celsius answer.

What if a problem uses the rounded offset 273?

Using K = C + 273 gives C = 301.25. The shared number becomes 574.25. Rounded to whole numbers, these are 301°C and 574 K, respectively.

Follow the conversion specified in the problem. Keep the requested scale clear before rounding.

Identify the quantity, write the unit, then calculate under the stated conditions. Keep dimensionless ratios separate from quantities with units.

• Convert prefixes before substituting values.
• Check whether a formula uses length, area or volume.
• Keep extra digits until the final rounding step.
• Check the requested temperature scale or instrument reading.

A light-year is a distance, not a time. It is the distance light travels in vacuum in one year.

Exam examples

CAPF (ACs) means Central Armed Police Forces (Assistant Commandants). NDA & NA means National Defence Academy and Naval Academy. CDS means Combined Defence Services.

• CAPF (ACs) 2026, Q113: relative density.
• NDA & NA (II) 2017, Q71, booklet A: henry and inductance.
• CDS (II) 2021, Q4: wire length and ammeter reading.
• NDA & NA (I) 2017, Q121: temperature-scale calculation.