Units, Dimensions and Measurement

Why Measurement Comes First

Physics is the science of measurement. Before you can talk about speed, force or energy, you must agree on how to measure length, mass and time. A measurement has two parts: a numerical value and a unit. Saying “the rope is 5” means nothing — 5 metres, 5 feet and 5 inches are all different.

In the NDA written exam this topic appears almost every year. The questions are short, formula-light and very predictable. If you master units and dimensions early, you also avoid silly mistakes in mechanics, optics and electricity later.

Think about how confusing the world would be without agreed units. A shopkeeper, an engineer and a soldier reading a map all need the same metre and the same kilogram, otherwise nothing would match. Over history people used many local systems — the foot, the seer, the mile — and this caused endless errors in trade and science. The whole point of a standard unit is that it is fixed, reproducible and the same everywhere, so that a measurement taken in Delhi means exactly the same in any other city or country.

Fundamental and Derived Quantities

Physical quantities are of two kinds. Fundamental (base) quantities are independent and cannot be defined using others. Derived quantities are built by combining fundamental ones.

• Fundamental: length, mass, time, electric current, temperature, amount of substance, luminous intensity.
• Derived: area (length × length), speed (length ÷ time), force (mass × acceleration), density, pressure, energy and so on.

There are exactly seven fundamental quantities. Everything else in physics is derived from these seven. The clever idea is that just seven independent building blocks are enough to describe every measurable quantity in nature, from the tiny mass of an electron to the vast distance between galaxies.

The SI System and Its Seven Base Units

The SI system (System International d'Unites) is the modern, world-wide standard. It has seven base units, and you must know all of them.

Two extra supplementary units are also defined: the radian (rad) for plane angle and the steradian (sr) for solid angle.

Many derived units get special names — newton (N) for force, joule (J) for energy, watt (W) for power, pascal (Pa) for pressure and hertz (Hz) for frequency. Each is still made from base units; for example 1 N = 1 kg·m·s⁻² and 1 J = 1 kg·m²·s⁻².

The SI system also uses standard prefixes to handle very large and very small numbers without writing long strings of zeros. Common ones to remember are kilo (10³), mega (10⁶) and giga (10⁹) for big quantities, and milli (10⁻³), micro (10⁻⁶) and nano (10⁻⁹) for small ones. So 1 km = 10³ m and 1 nm = 10⁻⁹ m. NDA sometimes tests these prefixes directly, so learn the powers of ten that go with each name.

You should also know the difference between SI and older systems. The CGS system uses centimetre, gram and second, while the FPS system uses foot, pound and second. SI is preferred today because it is coherent, decimal-based and used worldwide in science and the armed forces.

What Are Dimensions?

The dimensions of a quantity show how it depends on the base quantities. We use symbols in square brackets: [M] for mass, [L] for length and [T] for time (plus [A] for current, [K] for temperature).

The dimensional formula is the expression of a quantity in these symbols. For example:

• Area = length × length → [L²] or [M⁰L²T⁰]
• Speed = length ÷ time → [LT⁻¹]
• Acceleration = speed ÷ time → [LT⁻²]
• Force = mass × acceleration → [MLT⁻²]
• Work / Energy = force × distance → [ML²T⁻²]
• Power = energy ÷ time → [ML²T⁻³]

Dimensional Formulas You Must Memorise

NDA loves asking for the dimensional formula of one specific quantity. Keep this short list ready.

Notice that quantities with the same dimensions can be added or compared. Work, energy and torque all share [ML²T⁻²]. Stress and pressure share [ML⁻¹T⁻²]. The exam often tests these pairs by asking which two quantities have identical dimensions, so spotting these matches quickly saves valuable time.

A neat trick to build any dimensional formula is to start from its defining equation and replace each quantity by its own dimensions. For pressure, start from force ÷ area: [MLT⁻²] ÷ [L²] = [ML⁻¹T⁻²]. For the gravitational constant G, rearrange Newton's law F = Gm₁m₂÷r² to get G = Fr²÷m₁m₂, which gives [M⁻¹L³T⁻²]. Once you practise this method a few times, you can derive any formula on the spot rather than memorising a long list.

The Principle of Homogeneity

This single rule powers most dimensional-analysis questions. The principle of homogeneity states that every term added or equated in a physical equation must have the same dimensions.

You cannot add a length to a time, just as you cannot add metres to seconds. So in the equation s = ut + ½at², each term must reduce to [L]:

• ut → [LT⁻¹][T] = [L] ✓
• ½at² → [LT⁻²][T²] = [L] ✓

Both terms are [L], so the equation is dimensionally correct.

Uses and Limitations of Dimensional Analysis

Dimensional analysis is powerful but not magic. Know both sides for the exam.

What it can do

• Check whether an equation is dimensionally correct.
• Convert a unit from one system to another (e.g. CGS to SI).
• Derive a relation between quantities when the dependence is a simple power law.

What it cannot do

• Find dimensionless constants like ½ or 2π — these stay unknown.
• Handle equations involving sums, like s = ut + ½at², when deriving from scratch.
• Deal with quantities that depend on more than three base quantities at once using only three equations.
• Derive formulas with trigonometric, exponential or logarithmic functions.

Worked Example: Deriving a Formula

Let us use dimensional analysis to find how the time period of a simple pendulum depends on its length and gravity.

Mass drops out completely (a = 0), which matches the real result that a pendulum's period does not depend on its mass. The unknown constant k turns out to be 2π, but dimensional analysis cannot tell us that — it only gives the structure.

Errors in Measurement

No measurement is perfect. The difference between the measured value and the true value is the error. NDA tests three simple definitions.

Errors are of two broad types. Systematic errors repeat in the same direction (a faulty zero on a scale, a wrongly calibrated thermometer). Random errors vary unpredictably from reading to reading, caused by small changes in the observer, the surroundings or the instrument, and are reduced by taking many readings and averaging so that positive and negative errors cancel.

Two related ideas often confuse students. Accuracy tells you how close a measurement is to the true value, while precision tells you how closely repeated measurements agree with one another. A measurement can be precise but not accurate — for example a scale with a wrong zero gives very consistent but consistently wrong readings. Good experiments aim for both.

Combining errors

• When quantities are added or subtracted, their absolute errors add.
• When quantities are multiplied or divided, their relative (percentage) errors add.
• If a quantity is raised to a power n, its percentage error is multiplied by n.

Significant Figures and Rounding

Significant figures are the digits in a measurement that are reliably known plus one estimated digit. They tell us the precision of a reading.

Quick rules

• All non-zero digits are significant: 245 has 3.
• Zeros between non-zero digits count: 2004 has 4.
• Leading zeros do not count: 0.0025 has only 2.
• Trailing zeros after a decimal point count: 2.300 has 4.

In calculations

• In addition/subtraction, keep as many decimal places as the least precise number.
• In multiplication/division, keep as many significant figures as the number with the fewest.

Previous-Year Style Question

Quick Revision

This chapter is short and high-yield. Revise the dimensional formulas and error rules the night before the exam, and you can pick up these marks in seconds.