Units, Measurement and Dimensions

A quantity describes what is measured; a unit gives its reference scale. A dimension describes the quantity's dependence on base quantities.

• Quantity: a measurable property, such as length.
• Unit: an agreed reference, such as the metre.
• Dimension: the physical category represented by length, mass, time or other base quantities.
• Measurement: a numerical value expressed with an appropriate unit.

A table can be 2 m long or 200 cm long. The number and unit change together. The length itself and its dimension remain unchanged.

The International System of Units (SI) uses seven base quantities. Derived quantities combine these quantities through physical relationships.

Area combines two lengths, so its unit is m². Speed divides length by time, giving m/s.

Weight is a force, rather than a base quantity. Mass is the base quantity used when describing how much matter a body contains.

No. Scalar or vector character differs from dimensional character. A scalar needs a magnitude, while a vector also needs direction.

Speed and velocity both have dimensions of length divided by time. Speed is scalar; velocity is vector. Mass is scalar, while force is vector.

Pressure is also scalar. In a resting fluid, pressure forces act normally to surfaces. The force has direction, even though the pressure value is scalar.

Start from the quantity's defining relationship and replace each quantity with its dimensions. Mechanical formulas often use mass M, length L and time T.

Speed is distance divided by time, so [v] = LT⁻¹. Acceleration is velocity change divided by time, so [a] = LT⁻².

For constant mass, force satisfies F = ma. Therefore, [F] = M × LT⁻² = MLT⁻².

These three base dimensions cover the mechanical examples here. Electrical and thermal quantities can require other base dimensions.

A special unit name is shorthand for a combination of base units. Expand the name to understand the physical relationship.

Pressure is force divided by area. Dividing MLT⁻² by L² gives ML⁻¹T⁻².

Impulse has the same units as momentum change. A newton-second is not another name for a joule.

Longitudinal strain divides change in length by original length. Both lengths must use compatible units.

Suppose a 1 m wire lengthens by 2 mm. First convert 2 mm into 0.002 m.

strain = 0.002 m / 1 m = 0.002

The length units cancel, giving dimension 1, also written M⁰L⁰T⁰. Stress divides force by area, so it retains pressure dimensions and pascal units.

Relative density also compares two densities and is dimensionless. Dimensionless does not mean meaningless: the ratio still describes a physical comparison.

Some dimensionless quantities have named units, such as the radian for plane angle. Therefore, “dimensionless” need not mean that a unit label is never used.

Terms added together must have matching dimensions. Both sides of a physical equation must also match.

Consider the proposed distance equation s = ut + a. Here, u is initial speed and a is acceleration.

The term ut has dimension L. Acceleration alone has dimension LT⁻². Those terms cannot be added to produce a distance.

By contrast, at² has dimension L. Replacing a with at² removes the dimensional mismatch, but does not establish the correct numerical coefficient.

No. Dimensional consistency is necessary, but it is not sufficient. The equation must also represent the correct physical relationship.

Suppose a proposed energy depends on mass and speed as E = k mv². The product mv² has energy dimensions, so k must be dimensionless.

Dimensions alone cannot determine whether k is 1, ½ or 2. Measurements or physical reasoning must establish the coefficient and applicable conditions.

Dimensional analysis also cannot distinguish scalar speed from vector velocity solely through their dimensions. Check the meaning of the quantities as well.

An ångström equals 10⁻¹⁰ m; a fermi equals one femtometre, 10⁻¹⁵ m. Ångströms suit atomic-scale lengths, while femtometres suit nuclear-scale lengths.

Both measure length, despite describing very different scales. Changing the unit never changes the dimension from L.

A light-year measures the distance light travels in vacuum during one year. Time appears in its definition, but the resulting quantity is length.

The relationship is distance equals speed multiplied by time. Thus, LT⁻¹ × T = L.

A parsec is another astronomical distance unit. One parsec is approximately 206,265 astronomical units, or roughly 2 × 10⁵ au.

The astronomical unit, abbreviated au, is itself a unit of length. A parsec is larger than a light-year.

The kilogram is now defined through a fixed value of the Planck constant. The definition took effect in 2019.

The kilogram remains the SI unit of mass. Its definition no longer depends on the mass of a particular metal cylinder kept in France.

Changing the reference definition did not turn mass into a new kind of quantity. Its dimension remains M.

Resolution concerns small differences; accuracy concerns closeness to the true value. A fine scale can still have a zero error.

A vernier calliper combines a main scale with a sliding vernier scale. A screw gauge converts screw rotation into a small linear displacement.

School screw gauges commonly have a finer least count than school vernier callipers. Compare the actual instruments, rather than assuming every model has the same resolution.

For a screw gauge, least count = pitch / number of circular-scale divisions.

With pitch 1 mm and 100 divisions, least count is 0.01 mm. Pitch means the distance advanced in one complete rotation.

Check zero error and use consistent technique. Repeating measurements can reveal scatter, but repetition alone does not remove a systematic zero error.

The National Defence Academy and Naval Academy (NDA and NA) questions connect units with physical meaning.

Strain or stress? — NDA and NA I, 2025, Q139

Among stress, strain, pressure and force, strain is dimensionless. Stress and pressure have dimensions ML⁻¹T⁻². Force has dimensions MLT⁻².

Read the strain question.

How large is a parsec? — NDA and NA II, 2025, Q53

One parsec is about 2 × 10⁵ astronomical units. The alternatives 10³, 10⁴ and 10⁶ describe different scales.

Read the parsec question.

Is pressure a vector? — NDA and NA II, 2016, Q76

Pressure is scalar, while its resulting force has direction. For a resting fluid, the pressure force is normal to the surface.

Do not explain scalar pressure by claiming force and area are both vectors. The pressure value and the surface-force direction describe different parts of the relationship.

Read the pressure question.

What does a light-year measure? — NDA and NA II, 2017, Q110

A light-year measures distance. The word “year” describes the time used to define that distance, rather than the resulting quantity.

Read the light-year question.