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NDA Physics · Units, Measurement and Dimensions

Units, Measurement and Dimensions

Every physical quantity is measured against a unit; the seven SI base units build all derived units, and the dimensional formula [M^a L^b T^c] of a quantity lets you check equations, convert systems, and identify an unknown quantity from its units alone.

All Units, Measurement and Dimensions notesNDA Physics strategy

Why this matters

All 14 PYQs in NDA Physics's Units chapter live here, and about ten of them are EASY one-line recall. Light year as a unit of distance is asked four separate times; 1 dyne = 10⁻⁵ N, H = Henry, 1 kWh = 3.6×10⁶ J and strain-is-dimensionless are each near-certain marks. The only MODERATE/HARD work is one tool — the dimensional formula — used to find the dimension of G or to identify that thrust ÷ impulse is a frequency. Lock the recall tables and the dimensional recipe and the whole chapter becomes a lookup.

Concept 1 of 9

Physical quantities, units, and the seven SI base units

Intuition

A measurement is always a number times a unit: "5 metres" means 5 of the agreed length-standard. Some quantities are chosen as fundamental (base) and the rest are derived by multiplying and dividing them. The whole of mechanics rests on just three base units — kilogram (mass), metre (length), second (time) — and the SI system fixes seven base units in all.

Definition

A physical quantity is anything measurable; its measure is a numerical value × a unit.

  • Fundamental (base) quantities are independent and chosen by convention; in SI there are seven.
  • Derived quantities are built from the base ones (e.g. speed = length/time, force = mass × length/time²).
  • The three mechanics base units are kilogram (mass), metre (length), second (time) — these alone build every M–L–T dimensional formula.
Base quantitySI unitSymbol
Lengthmetrem
Masskilogramkg
Timeseconds
Electric currentampereA
TemperaturekelvinKQ
NDA 2025 match-list — Temperature → Kelvin, Mass → Kilogram (weight is a force → Newton, pressure → Pascal).
Amount of substancemolemol
Luminous intensitycandelacd
The seven SI base units. Mass is the kilogram; weight is a force (newton), not a base unit — the classic match-list trap.
Practice this conceptself-check · 4 quick reps

Try it yourself

Match each quantity to its SI base unit: temperature, mass, electric current, amount of substance.

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

  1. 1.
    SI base unit of temperature?
  2. 2.
    SI base unit of mass?
  3. 3.
    SI base unit of electric current?
  4. 4.
    How many SI base units are there?

From the bank · past-year question

Example 1Units, Measurement and DimensionsEASY
Match List I with List II and select the answer using the code given below the Lists : A. Temperature | 1. Kelvin B. Weight | 2. Kilogram C. Mass | 3. Pascal D. Pressure | 4. Newton

[Q84 · Apr · 2025]

Mass is kilogram; weight is a force (newton)

In a match-list, "Weight" pairs with Newton, not kilogram — weight is the gravitational force mgmg, a derived unit. Only mass maps to the kilogram base unit. Pressure maps to the pascal, temperature to the kelvin.

Concept 2 of 9

SI derived units named after scientists

Intuition

Many derived units carry a scientist's name and an agreed symbol — newton (force), pascal (pressure), joule (energy), watt (power), hertz (frequency), henry (inductance). The exam tests two things: what a symbol stands for (H = Henry, not Hertz) and which two quantities share a unit (stress and pressure are both N/m²).

Definition

Common named SI derived units:

  • Newton (N) — force; 1 N=1 kg⋅m/s21\,\text{N} = 1\,\text{kg·m/s}^2.
  • Pascal (Pa) — pressure and stress; 1 Pa=1 N/m21\,\text{Pa} = 1\,\text{N/m}^2.
  • Joule (J) — work and energy; 1 J=1 N⋅m1\,\text{J} = 1\,\text{N·m}.
  • Watt (W) — power; 1 W=1 J/s1\,\text{W} = 1\,\text{J/s}.
  • Hertz (Hz) — frequency; 1 Hz=1 s−11\,\text{Hz} = 1\,\text{s}^{-1}.
  • Henry (H) — inductance.

Stress and pressure are both force ÷ area, so they share the unit N/m² (pascal).

Unit (symbol)QuantityIn base units
Newton (N)Forcekg·m/s²
Pascal (Pa)Pressure, stressN/m² = kg/(m·s²)
Joule (J)Work, energyN·m = kg·m²/s²
Watt (W)PowerJ/s = kg·m²/s³
Hertz (Hz)Frequencys⁻¹
Henry (H)Inductancekg·m²/(s²·A²)Q
NDA 2017 — the symbol H stands for Henry (after Joseph Henry), NOT Hertz.
Stress and pressure share the same unit (N/m²). The symbol H is Henry (inductance); Hz is the hertz (frequency).
Practice this conceptself-check · 4 quick reps

Try it yourself

Which physical quantity has the same SI unit as pressure: angular momentum, stress, strain, or work?

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

  1. 1.
    What does the SI symbol H stand for?
  2. 2.
    Which quantity shares pressure's unit?
  3. 3.
    SI unit of power?
  4. 4.
    SI unit of frequency?

From the bank · past-year question

Example 2Units, Measurement and DimensionsEASY
Which one of the following physical quantity has the same unit as that of pressure ?

[Q108 · Apr · 2017]

H is Henry, not Hertz

The symbol H is the henry (unit of inductance, after Joseph Henry). Hertz has the symbol Hz and measures frequency. The exam offers both as distractors.

Stress and pressure share a unit

Both stress and pressure are force per unit area (N/m² = pascal). Strain, by contrast, is a pure ratio and is dimensionless — don't confuse stress (has a unit) with strain (no unit).
Drill 1 more on si derived units named after scientists

Concept 3 of 9

Units of length and distance — light year, ångström, nanometre

Intuition

Lengths span an enormous range, so physics uses special units at each end. For astronomical DISTANCES we use the light year and the parsec; for atomic-scale lengths we use the ångström and the nanometre. The single most-tested fact in this whole chapter: a light year is a unit of distance, not time and not intensity — it has been asked four separate times.

Definition

Special length units, large to small:

  • Light year (ly) — the distance light travels in one year, about 9.46×10159.46 \times 10^{15} m. It is a unit of distance only, never time or light intensity.
  • Astronomical unit (AU) — mean Earth–Sun distance, about 1.496×10111.496 \times 10^{11} m.
  • Parsec (pc) — about 3.263.26 light years (3.086×10163.086 \times 10^{16} m), the astronomer's distance unit.
  • Nanometre (nm) =10−9= 10^{-9} m.
  • Ångström (Å) =10−10= 10^{-10} m — so 1 nm=10 A˚1\,\text{nm} = 10\,\text{Å}.
UnitMeasuresValue
Light year (ly)Distance (astronomical)9.46 × 10¹⁵ mQ
Asked 4× (2017, 2018, 2021) — light year is DISTANCE, never time, never light intensity.
Astronomical unit (AU)Distance (Earth–Sun)1.496 × 10¹¹ m
Parsec (pc)Distance (astronomical)3.086 × 10¹⁶ m ≈ 3.26 ly
Nanometre (nm)Length (atomic-scale)10⁻⁹ m
Ångström (Å)Length (atomic-scale)10⁻¹⁰ mQ
NDA 2018 — 1 nm = 10 Å (since nm is 10⁻⁹ m and Å is 10⁻¹⁰ m).
Light year, AU and parsec all measure DISTANCE. 1 nm = 10 Å. The light-year-is-distance fact is the chapter's single highest-yield line.
Practice this conceptself-check · 4 quick reps

Try it yourself

Consider these statements about a light year: (1) it measures very large distances, (2) it measures very large time intervals, (3) it measures light intensity. Which are correct?

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

  1. 1.
    A light year measures what?
  2. 2.
    1 nm equals how many ångströms?
  3. 3.
    1 ångström in metres?
  4. 4.
    Is the light year a unit of time?

From the bank · past-year question

Example 3Units, Measurement and DimensionsEASY
Consider the following statements about Light year: 1. Light year is a unit for measurement of very large distances. 2. Light year is a unit for measurement of very large time intervals. 3. Light year is a unit for measurement of intensity of light. Which of the statements given above is/are correct ?

[Q51 · Apr · 2021]

Light year is DISTANCE, not time

The word "year" plants the trap: a light year is the distance light covers in a year (~9.46×10159.46 \times 10^{15} m), not a time interval and not light intensity. This exact fact is asked again and again.

1 nm = 10 Å (not 0.1 Å)

Since 1 nm =10−9= 10^{-9} m and 1 Å =10−10= 10^{-10} m, the nanometre is the larger unit: 1 nm=10 A˚1\,\text{nm} = 10\,\text{Å}. Don't invert it.
Drill 3 more on units of length and distance — light year, ångström, nanometre

Concept 4 of 9

Units of energy and power — joule, kWh, and the force trap

Intuition

Energy and work share the joule; the commercial unit of electrical energy is the kilowatt-hour (kWh). The recurring trap hands you a list of energy units and slips in a force unit — kg·m/s² is a newton (force), not energy. Any newton-metre (N·m) is a joule, and watt-hour is power × time = energy.

Definition

Energy / work is measured in the joule (J): 1 J=1 N⋅m=1 kg⋅m2/s21\,\text{J} = 1\,\text{N·m} = 1\,\text{kg·m}^2/\text{s}^2, with dimension [ML2T−2][ML^2T^{-2}]. The kilowatt-hour (kWh) is energy = power × time: 1 kWh=1000 W×3600 s=3.6×106 J1\,\text{kWh} = 1000\,\text{W} \times 3600\,\text{s} = 3.6 \times 10^{6}\,\text{J}. Watt-hour and newton-metre are also energy units; kg·m/s² is the newton — a unit of force, not energy.

Kilowatt-hour to joules

1 kWh=1000 W×3600 s=3.6×106 J1\,\text{kWh} = 1000\,\text{W} \times 3600\,\text{s} = 3.6 \times 10^{6}\,\text{J}
  • Wwatt = joule per second (power)
  • kWhkilowatt-hour, the commercial unit of electrical energy

Worked example

An electric heater rated at 2 kW runs for 3 hours. How much energy does it consume, in kWh and in joules?
  1. Energy in kWh = power × time = 2 kW×3 h=6 kWh2\,\text{kW} \times 3\,\text{h} = 6\,\text{kWh}.
  2. Convert one kWh: 1 kWh=1000×3600=3.6×106 J1\,\text{kWh} = 1000 \times 3600 = 3.6 \times 10^{6}\,\text{J}.
  3. So 6 kWh=6×3.6×106=2.16×107 J6\,\text{kWh} = 6 \times 3.6 \times 10^{6} = 2.16 \times 10^{7}\,\text{J}.
Answer:6 kWh, which is 2.16×1072.16 \times 10^{7} J.
Practice this conceptself-check · 4 quick reps

Try it yourself

Which of these is NOT a unit of energy: joule, watt-hour, newton-metre, or kg·m/s²?

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

  1. 1.
    1 kWh in joules?
  2. 2.
    Is newton-metre a unit of energy?
  3. 3.
    Is kg·m/s² a unit of energy?
  4. 4.
    Energy used by a 100 W bulb in 10 h, in kWh?

From the bank · past-year question

Example 4Units, Measurement and DimensionsEASY
Which one of the following is the value of 1 kWh of energy converted into joules?

[Q57 · Apr · 2018]

kg·m/s² is force, not energy

In a "which is NOT a unit of energy" list, the planted answer is kg·m/s² — that is mass × acceleration = the newton (force). Energy is kg·m²/s² (joule). Watch the exponent on the metre.

1 kWh = 3.6 × 10⁶ J, not 3600

It's 1000 W × 3600 s = 3.6×1063.6 \times 10^{6} J. Multiplying only by 3600 (forgetting the kilo) gives 3.6×1033.6 \times 10^{3} — a factor-of-1000 error.
Drill 1 more on units of energy and power — joule, kwh, and the force trap

Concept 5 of 9

Unit-system conversion — CGS to SI (the dyne)

Intuition

The CGS system uses centimetre, gram, second; SI uses metre, kilogram, second. To convert a unit, substitute the base-unit factors: 1 g = 10⁻³ kg and 1 cm = 10⁻² m. Doing this for force gives the headline conversion: 1 dyne = 10⁻⁵ newton.

Definition

Convert a unit by replacing each base unit with its SI value:

  • 1 g=10−3 kg1\,\text{g} = 10^{-3}\,\text{kg}, 1 cm=10−2 m1\,\text{cm} = 10^{-2}\,\text{m}.
  • Force: 1 dyne=1 g⋅cm/s2=10−3 kg×10−2 m/s2=10−5 N1\,\text{dyne} = 1\,\text{g·cm/s}^2 = 10^{-3}\,\text{kg} \times 10^{-2}\,\text{m/s}^2 = 10^{-5}\,\text{N}.
  • Energy: 1 erg=1 g⋅cm2/s2=10−7 J1\,\text{erg} = 1\,\text{g·cm}^2/\text{s}^2 = 10^{-7}\,\text{J}.

CGS force unit to SI

1 dyne=1 g⋅cms2=(10−3 kg)(10−2 m) s−2=10−5 N1\,\text{dyne} = 1\,\frac{\text{g·cm}}{\text{s}^2} = (10^{-3}\,\text{kg})(10^{-2}\,\text{m})\,\text{s}^{-2} = 10^{-5}\,\text{N}
  • dyneCGS unit of force (g·cm/s²)
  • NSI unit of force (kg·m/s²)

Worked example

Convert 1 erg (the CGS unit of energy, g·cm²/s²) into joules.
  1. Write erg in base units: 1 erg=1 g⋅cm2/s21\,\text{erg} = 1\,\text{g·cm}^2/\text{s}^2.
  2. Substitute 1 g=10−3 kg1\,\text{g} = 10^{-3}\,\text{kg} and (1 cm)2=(10−2 m)2=10−4 m2(1\,\text{cm})^2 = (10^{-2}\,\text{m})^2 = 10^{-4}\,\text{m}^2.
  3. 1 erg=10−3×10−4 kg⋅m2/s2=10−7 J1\,\text{erg} = 10^{-3} \times 10^{-4}\,\text{kg·m}^2/\text{s}^2 = 10^{-7}\,\text{J}.
Answer:1 erg=10−71\,\text{erg} = 10^{-7} J.
Practice this conceptself-check · 4 quick reps

Try it yourself

Express 1 dyne in SI base units.

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

  1. 1.
    1 dyne in newtons?
  2. 2.
    1 erg in joules?
  3. 3.
    1 gram in kilograms?
  4. 4.
    1 newton in dynes?

From the bank · past-year question

Example 5Units, Measurement and DimensionsMODERATE
1 dyne (a unit of force in CGS system) equals to

[Q113 · Apr · 2019]

1 dyne = 10⁻⁵ N (not 10⁻³ N)

Both factors shrink the unit: gram → kg gives 10−310^{-3} and cm → m gives 10−210^{-2}. Multiply them: 10−3×10−2=10−510^{-3} \times 10^{-2} = 10^{-5}. Forgetting the centimetre factor gives the wrong 10−3 N10^{-3}\,\text{N} distractor.

Concept 6 of 9

Dimensional formulas — writing [M^a L^b T^c]

Intuition

The dimensional formula of a quantity records how it is built from mass (M), length (L) and time (T). Write the defining equation, replace each quantity by its dimensions, and simplify the powers. Solving the gravitation law for G gives its dimension; the same recipe checks any formula for consistency.

Definition

The dimensional formula expresses a quantity as [MaLbTc][M^a L^b T^c] (with current, temperature etc. added when needed).

  • Velocity [LT−1][LT^{-1}], acceleration [LT−2][LT^{-2}], force [MLT−2][MLT^{-2}], energy/work [ML2T−2][ML^2T^{-2}], pressure [ML−1T−2][ML^{-1}T^{-2}].
  • Method: start from a defining equation, substitute dimensions for every symbol, and collect the powers of M, L, T.
  • A correct physical equation must be dimensionally homogeneous — both sides carry the same dimensions.

Dimension of the gravitational constant G

F=Gm1m2r2  ⇒  [G]=[F][r2][m2]=(MLT−2)(L2)M2=M−1L3T−2F = \frac{G m_1 m_2}{r^2} \;\Rightarrow\; [G] = \frac{[F][r^2]}{[m^2]} = \frac{(MLT^{-2})(L^2)}{M^2} = M^{-1}L^3T^{-2}
  • Fgravitational force, [MLT⁻²]
  • m_1, m_2masses, [M] each
  • rseparation, [L]

Worked example

Find the dimensional formula of pressure.
  1. Pressure = force ÷ area.
  2. Force has dimension [MLT−2][MLT^{-2}]; area has dimension [L2][L^2].
  3. [P]=MLT−2L2=ML−1T−2[P] = \dfrac{MLT^{-2}}{L^2} = ML^{-1}T^{-2}.
Answer:[P]=ML−1T−2[P] = M L^{-1} T^{-2}.
Practice this conceptself-check · 4 quick reps

Try it yourself

Derive the dimensional formula of the gravitational constant G from Newton's law of gravitation.

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

  1. 1.
    Dimensional formula of force?
  2. 2.
    Dimensional formula of energy?
  3. 3.
    Dimensional formula of G?
  4. 4.
    Dimensional formula of velocity?

From the bank · past-year question

Example 6Units, Measurement and DimensionsMODERATE
What is the dimension of gravitational constant?

[Q68 · Apr · 2022]

G carries a NEGATIVE mass power: M⁻¹

Because G = F·r²/(m₁m₂), the two masses sit in the denominator, giving M−1M^{-1} — not M+1M^{+1}. The full result M−1L3T−2M^{-1}L^3T^{-2} has L3L^3 (from force's L times r²'s L²) and T−2T^{-2} (from force). A sign slip on M is the planted error.

Concept 7 of 9

Dimensionless quantities — strain, angle, refractive index

Intuition

A quantity built as a ratio of two same-dimension quantities has all its dimensions cancel — it is dimensionless and has no unit. Strain (change in length ÷ original length), refractive index (ratio of speeds), relative density and plane angle (arc ÷ radius) are the standard dimensionless quantities the exam tests.

Definition

A dimensionless quantity has dimensional formula [M0L0T0][M^0 L^0 T^0] — no unit.

  • Strain = change in length ÷ original length = L/LL/L → dimensionless. (Contrast stress = force/area, which has units N/m².)
  • Refractive index = speed in vacuum ÷ speed in medium → dimensionless.
  • Relative density (specific gravity) = density ÷ density of water → dimensionless.
  • Plane angle (radian) = arc length ÷ radius = L/LL/L → dimensionless.

Strain is a pure ratio

strain=ΔLL=[L][L]=[M0L0T0]\text{strain} = \frac{\Delta L}{L} = \frac{[L]}{[L]} = [M^0 L^0 T^0]
  • \Delta Lchange in length, [L]
  • Loriginal length, [L]

Worked example

Among stress, strain, pressure and force, which one is dimensionless?
  1. Stress = force/area = [ML−1T−2][ML^{-1}T^{-2}] — has dimensions.
  2. Pressure = force/area = [ML−1T−2][ML^{-1}T^{-2}] — has dimensions.
  3. Force = [MLT−2][MLT^{-2}] — has dimensions.
  4. Strain = ΔL/L, a length-over-length ratio → [M0L0T0][M^0L^0T^0], dimensionless.
Answer:Strain is the dimensionless quantity.
Practice this conceptself-check · 4 quick reps

Try it yourself

Is the refractive index of glass a dimensionless quantity? Justify in one line.

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

  1. 1.
    Is strain dimensionless?
  2. 2.
    Is stress dimensionless?
  3. 3.
    Is refractive index dimensionless?
  4. 4.
    Is plane angle (radian) dimensionless?

From the bank · past-year question

Example 7Units, Measurement and DimensionsEASY
Which one of the following is dimensionless quantity?

[Q139 · Apr · 2025]

Strain is dimensionless; stress is NOT

Strain is the ratio ΔL/L (no unit). Stress is force ÷ area (N/m²). The pair is designed to be confused — only strain is dimensionless.

Concept 8 of 9

Identifying a quantity from its units or dimensions

Intuition

When asked what an unfamiliar combination 'is the same as', reduce both candidate and combination to base units (or M–L–T dimensions) and compare. Thrust is a force and impulse is force × time, so thrust ÷ impulse is 1/time — which is a frequency (hertz).

Definition

To identify an unknown combination, reduce it to base units / dimensions and match it to a known quantity.

  • Thrust = force, [MLT−2][MLT^{-2}] (unit N).
  • Impulse = force × time = change in momentum, [MLT−1][MLT^{-1}] (unit N·s).
  • So thrust ÷ impulse =MLT−2MLT−1=T−1= \dfrac{MLT^{-2}}{MLT^{-1}} = T^{-1} → the dimension of frequency (unit Hz).

Thrust ÷ impulse is a frequency

thrustimpulse=[MLT−2][MLT−1]=[T−1]=frequency (Hz)\frac{\text{thrust}}{\text{impulse}} = \frac{[MLT^{-2}]}{[MLT^{-1}]} = [T^{-1}] = \text{frequency (Hz)}
  • thrusta force, [MLT⁻²]
  • impulseforce × time, [MLT⁻¹]

Worked example

The unit of (force ÷ momentum) is the same as that of which quantity?
  1. Force has dimension [MLT−2][MLT^{-2}].
  2. Momentum = mass × velocity has dimension [MLT−1][MLT^{-1}].
  3. Force ÷ momentum =MLT−2/MLT−1=T−1= MLT^{-2} / MLT^{-1} = T^{-1} — the dimension of frequency.
Answer:Frequency (it has dimension T−1T^{-1}, unit Hz).
Practice this conceptself-check · 4 quick reps

Try it yourself

The ratio of thrust to impulse has the same unit as which quantity — speed, wavelength, acceleration, or frequency?

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

  1. 1.
    Dimension of thrust ÷ impulse?
  2. 2.
    Dimension of impulse?
  3. 3.
    Which quantity has dimension T⁻¹?
  4. 4.
    Force ÷ momentum has the unit of?

From the bank · past-year question

Example 8Units, Measurement and DimensionsHARD
The unit of the ratio between thrust and impulse is same as that of

[Q145 · Sep · 2021]

Impulse is force × TIME, not force

Impulse = F·t = change in momentum [MLT−1][MLT^{-1}], one power of T less negative than force [MLT−2][MLT^{-2}]. The thrust/impulse ratio therefore leaves T−1T^{-1} (frequency). Treating impulse as a plain force would wrongly make the ratio dimensionless.

Concept 9 of 9

Measurement — precision, accuracy and least count

Intuition

The least count is the smallest division an instrument can read — 1 mm for an ordinary metre scale. A measurement is only as precise as the instrument's least count; recording a value to a finer level than the least count is false precision. Among several readings, the one written consistently with the instrument's least count (here, to the millimetre) is the most precise honest measurement.

Definition

  • Least count (LC) — the smallest value an instrument can measure (metre scale: 1 mm; vernier callipers: 0.1 mm; screw gauge: 0.01 mm).
  • Precision — how finely a result is recorded; it is limited by the least count.
  • Accuracy — how close a result is to the true value (a separate idea from precision).
  • A metre scale with LC = 1 mm can sensibly report a length to the millimetre: e.g. 910 mm is read at exactly the scale's precision, whereas 0.925 m (to the mm via metres) or 29.07 cm imply a finer reading than the scale allows.

Least count of a metre scale

LC (metre scale)=1 mm=0.1 cm=10−3 m\text{LC (metre scale)} = 1\,\text{mm} = 0.1\,\text{cm} = 10^{-3}\,\text{m}
  • LCleast count — smallest readable division
0123456cm1 mm = least count

Each centimetre is split into ten 1 mm divisions. The smallest readable division — 1 mm — is the least count, so an ordinary metre scale reports a length honestly only to the nearest millimetre.

Worked example

A metre scale has a least count of 1 mm. A student records a rod's length as 24.6 cm. To what precision is this honest, and how would you write it in mm?
  1. Least count 1 mm = 0.1 cm, so the scale can resolve to the first decimal of a centimetre.
  2. 24.6 cm is given to 0.1 cm = 1 mm — exactly the scale's precision, so it is an honest reading.
  3. In millimetres: 24.6 cm=246 mm24.6\,\text{cm} = 246\,\text{mm}.
Answer:It is precise to 1 mm (the least count); the length is 246 mm.
Practice this conceptself-check · 4 quick reps

Try it yourself

Using a metre scale of least count 1 mm, which reading is recorded most consistently with the instrument's precision: 0.50 mm, 29.07 cm, 0.925 m, or 910 mm?

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

  1. 1.
    Least count of an ordinary metre scale?
  2. 2.
    Least count of a vernier calliper?
  3. 3.
    Does a smaller least count mean greater precision?
  4. 4.
    Is precision the same as accuracy?

From the bank · past-year question

Example 9Units, Measurement and DimensionsMODERATE
A student measures certain lengths using a meter scale having least count equal to 1 mm. Which one of the following measurements is more precise?

[Q68 · Sep · 2019]

Precision is set by the least count

A metre scale (LC = 1 mm) cannot honestly report below 1 mm. A value written to sub-millimetre detail (0.925 m, 29.07 cm) claims more precision than the instrument has; the reading recorded to the millimetre is the consistent one.

Precision ≠ accuracy

Precision is how finely you can read (least count); accuracy is how close you are to the true value. A finely-recorded reading can still be inaccurate, and vice versa — don't equate the two.

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (6)

  • Units of energy and power — joule, kWh, and the force trap

    Kilowatt-hour to joules

    1 kWh=1000 W×3600 s=3.6×106 J1\,\text{kWh} = 1000\,\text{W} \times 3600\,\text{s} = 3.6 \times 10^{6}\,\text{J}
  • Unit-system conversion — CGS to SI (the dyne)

    CGS force unit to SI

    1 dyne=1 g⋅cms2=(10−3 kg)(10−2 m) s−2=10−5 N1\,\text{dyne} = 1\,\frac{\text{g·cm}}{\text{s}^2} = (10^{-3}\,\text{kg})(10^{-2}\,\text{m})\,\text{s}^{-2} = 10^{-5}\,\text{N}
  • Dimensional formulas — writing [M^a L^b T^c]

    Dimension of the gravitational constant G

    F=Gm1m2r2  ⇒  [G]=[F][r2][m2]=(MLT−2)(L2)M2=M−1L3T−2F = \frac{G m_1 m_2}{r^2} \;\Rightarrow\; [G] = \frac{[F][r^2]}{[m^2]} = \frac{(MLT^{-2})(L^2)}{M^2} = M^{-1}L^3T^{-2}
  • Dimensionless quantities — strain, angle, refractive index

    Strain is a pure ratio

    strain=ΔLL=[L][L]=[M0L0T0]\text{strain} = \frac{\Delta L}{L} = \frac{[L]}{[L]} = [M^0 L^0 T^0]
  • Identifying a quantity from its units or dimensions

    Thrust ÷ impulse is a frequency

    thrustimpulse=[MLT−2][MLT−1]=[T−1]=frequency (Hz)\frac{\text{thrust}}{\text{impulse}} = \frac{[MLT^{-2}]}{[MLT^{-1}]} = [T^{-1}] = \text{frequency (Hz)}
  • Measurement — precision, accuracy and least count

    Least count of a metre scale

    LC (metre scale)=1 mm=0.1 cm=10−3 m\text{LC (metre scale)} = 1\,\text{mm} = 0.1\,\text{cm} = 10^{-3}\,\text{m}

Reference tables (3)

Physical quantities, units, and the seven SI base units7 rows
Base quantitySI unitSymbol
Lengthmetrem
Masskilogramkg
Timeseconds
Electric currentampereA
TemperaturekelvinKQ
NDA 2025 match-list — Temperature → Kelvin, Mass → Kilogram (weight is a force → Newton, pressure → Pascal).
Amount of substancemolemol
Luminous intensitycandelacd
The seven SI base units. Mass is the kilogram; weight is a force (newton), not a base unit — the classic match-list trap.
SI derived units named after scientists6 rows
Unit (symbol)QuantityIn base units
Newton (N)Forcekg·m/s²
Pascal (Pa)Pressure, stressN/m² = kg/(m·s²)
Joule (J)Work, energyN·m = kg·m²/s²
Watt (W)PowerJ/s = kg·m²/s³
Hertz (Hz)Frequencys⁻¹
Henry (H)Inductancekg·m²/(s²·A²)Q
NDA 2017 — the symbol H stands for Henry (after Joseph Henry), NOT Hertz.
Stress and pressure share the same unit (N/m²). The symbol H is Henry (inductance); Hz is the hertz (frequency).
Units of length and distance — light year, ångström, nanometre5 rows
UnitMeasuresValue
Light year (ly)Distance (astronomical)9.46 × 10¹⁵ mQ
Asked 4× (2017, 2018, 2021) — light year is DISTANCE, never time, never light intensity.
Astronomical unit (AU)Distance (Earth–Sun)1.496 × 10¹¹ m
Parsec (pc)Distance (astronomical)3.086 × 10¹⁶ m ≈ 3.26 ly
Nanometre (nm)Length (atomic-scale)10⁻⁹ m
Ångström (Å)Length (atomic-scale)10⁻¹⁰ mQ
NDA 2018 — 1 nm = 10 Å (since nm is 10⁻⁹ m and Å is 10⁻¹⁰ m).
Light year, AU and parsec all measure DISTANCE. 1 nm = 10 Å. The light-year-is-distance fact is the chapter's single highest-yield line.

Watch out for (13)

Mastery check — 5 interleaved questions

Try each one before clicking. Questions are interleaved across the concepts above, not grouped — interleaving sharpens transfer.

Example 1Units, Measurement and DimensionsEASY
The symbol of SI unit of inductance is H. It stands for

[Q71 · Sep · 2017]

Example 2Units, Measurement and DimensionsEASY
Light year is a unit for measurement of

[Q118 · Apr · 2018]

Example 3Units, Measurement and DimensionsEASY
Which one of the following is NOT the unit of energy?

[Q123 · Apr · 2020]

Example 4Units, Measurement and DimensionsEASY
Light year is a measure of

[Q110 · Sep · 2017]

Example 5Units, Measurement and DimensionsEASY
Which one of the following is the correct relation between Å and nm?

[Q70 · Sep · 2018]

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