Units and Measurements

A physical quantity P is related to four observations a, b, c and d as follows: P = a³b²/(c√d) The percentage errors of measurement in a, b, c and d are 1%, 3%, 2%, and 4% respectively. The percentage error in the quantity P is

NEET 2025

115%

210%

32%

413%

NTA Answer: Option 4(final)

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The quantities which have the same dimensions as those of solid angle are :

NEET 2024Revised key

1strain and angle

2stress and angle

3strain and arc

4angular speed and stress

NTA Answer: Option 1(revised_final)

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A force defined by F = αt2 + βt acts on a particle at a given time t. The factor which is dimensionless, if α and β are constants, is: βt

NEET 2024Revised key

1α

2αt β

3αβt αβ

NTA Answer: Option 2(revised_final)

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The errors in the measurement which arise due to unpredictable fluctuations in temperature and voltage supply are

NEET 2023

1Personal errors

2Least count errors

3Random errors

4Instrumental errors

NTA Answer: Option 3(final)

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A metal wire has mass (0.4 ± 0.002) g, radius (0.3 ± 0.001) mm and length (5 ± 0.02) cm. The maximum possible percentage error in the measurement of density will nearly be

NEET 2023

11.3%

21.6%

31.4%

41.2%

NTA Answer: Option 2(final)

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The dimensions [MLT⁻²A⁻²] belong to the

NEET 2022

1Electric permittivity

2Magnetic flux

3Self inductance

4Magnetic permeability

NTA Answer: Option 4(final)

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Lesson: Units of Measurements

NEET 2022

Plane angle and solid angle have

1Both units and dimensions

2Units but no dimensions

3Dimensions but no units

4No units and no dimensions

NTA Answer: Option 2(final)

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Match List-I with List-II List-I List-II (a) Gravitational constant (G) (i) [L²T⁻²] (b) Gravitational potential energy (ii) [M⁻¹L³T⁻²] (c) Gravitational potential (iii) [LT⁻²] (d) Gravitational intensity (iv) [ML²T⁻²] Choose the correct answer from the options given below

NEET 2022

1(a) - (iv), (b) - (ii), (c) - (i), (d) - (iii)

2(a) - (ii), (b) - (i), (c) - (iv), (d) - (iii)

3(a) - (ii), (b) - (iv), (c) - (i), (d) - (iii)

4(a) - (ii), (b) - (iv), (c) - (iii), (d) - (i)

NTA Answer: Option 3(final)

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The area of a rectangular field (in m²) of length 55.3 m and breadth 25 m after rounding off the value for correct significant digits is

NEET 2022

114 × 10²

2138 × 10¹

31382

41382.5

NTA Answer: Option 1(final)

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NEET 2021

A screw gauge gives the following readings when used to measure the diameter of a wire Main scale reading : 0 mm Circular scale reading : 52 divisions Given that 1 mm on main scale corresponds to 100 divisions on the circular scale. The diameter of the wire from the above data is

10.052 cm

20.52 cm

30.026 cm

40.26 cm

NTA Answer: Option 1(final)

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Taking into account of the significant figures, what is the value of 9.99 m−0.0099 m ?

NEET 2020

19.9 m

29.9801 m

39.98 m

49.980 m

NTA Answer: Option 3(final)

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NEET 2020

A screw gauge has least count of 0.01 mm and there are 50 divisions in its circular scale. The pitch of the screw gauge is :

11.0 mm

20.01 mm

30.25 mm

40.5 mm

NTA Answer: Option 4(final)

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Exam traps and common mistakes in this unit

Trap

Dimensional analysis time-exponent off by one

Category: Similar Terms

Student gets the time exponent wrong by 1 (e.g. T⁻¹ vs T⁻²) when manipulating dimensional formulas.

When it triggers

Question asks for dimensions of a derived combination (e.g. E/G, F = αt² + βt) where time exponent matters.

How to avoid

Write each base quantity's dimensional formula explicitly, then combine. Common errors: dividing forces forgets sub of T exponents; energy/length includes implicit time. Always check final units against expected SI.

Trap

Plane/solid angle: unit vs dimension confusion

Category: Similar Terms

Student treats radian/steradian as having dimensions because they have unit names.

When it triggers

Question asks about dimensions of plane angle, solid angle, or comparison.

How to avoid

Plane angle (radian) and solid angle (steradian) are DIMENSIONLESS — they're ratios (arc/radius for radian; surface-area/r² for steradian). They have unit NAMES for clarity but no dimensions.

Trap

Error propagation: missing power factor

Category: Similar Terms

Student sums relative errors of all measured quantities without weighting by the exponent. For ρ = m/(πr²L), the relative error contribution of r is 2 × Δr/r, NOT Δr/r — the exponent of r in the formula carries through as a multiplicative factor.

When it triggers

Question gives a derived quantity formula with mixed-power dependencies; asks for the max relative error. Distractors omit the power factor.

How to avoid

Always write the full general rule: for Z = A^p B^q C^r, ΔZ/Z = |p|·ΔA/A + |q|·ΔB/B + |r|·ΔC/C. Identify the powers (1, 2, 3, ½) before adding.

Trap

Error taxonomy: random vs instrumental vs systematic

Category: Similar Terms

Student conflates random errors (statistical, unpredictable, reduced by averaging) with instrumental errors (consistent bias from the apparatus) or with systematic errors (consistent bias from the method). Each has a distinct definition and different mitigation.

When it triggers

Question describes an error source and asks for its taxonomic category. Distractors include cognate categories.

How to avoid

Memorise the 5-category taxonomy: PERSONAL (observer-side), INSTRUMENTAL (apparatus calibration), LEAST-COUNT (instrument resolution floor), RANDOM (statistical, reduced by repeated trials), SYSTEMATIC (method-level bias, NOT reduced by averaging).

Trap

Significant-figures rule swap (multiplication rule applied to addition/subtraction)

Category: Similar Terms

Student applies the 'fewest significant figures' rule (which governs multiplication and division) to a sum or difference. Subtraction of two measured numbers must instead reflect the FEWEST decimal places.

When it triggers

Question involves addition/subtraction of measured numbers with very different magnitudes or decimal-place counts (e.g. 9.99 - 0.0099). Distractors offer answers rounded by sig-fig rule rather than decimal-place rule.

How to avoid

Memorise: multiplication/division → fewest SIGNIFICANT FIGURES; addition/subtraction → fewest DECIMAL PLACES. Always identify which arithmetic operation is being performed before applying any rule.

Unit-wide

Trap

Screw gauge — pitch vs least count confusion

Category: Similar Terms

Student swaps which is the input vs the output: least count = pitch / N (where N is the number of circular-scale divisions). Distractors offer the ratio inverted or the wrong unit.

When it triggers

Question gives one of (pitch, N, least count) and asks for another; distractors offer the inverted ratio or off-by-factor-of-10.

How to avoid

Anchor on the definition: least count is the SMALLEST measurement the instrument can resolve. It is always SMALLER than the pitch. So pitch = LC × N (and not LC = pitch × N).

Unit-wide

Trap

Vernier division-count inversion

Category: Similar Terms

Confusing whether N or N+1 is the smaller count when (N+1) divisions of vernier match N divisions of main scale.

When it triggers

Question gives '(N+1) divisions of vernier coincide with N divisions of main' or similar phrasing.

How to avoid

Always interpret carefully: N+1 vernier divisions span the SAME LENGTH as N main divisions. So 1 VSD = (N/(N+1)) MSD; vernier constant = 1 MSD - 1 VSD = 1 MSD / (N+1). Result smaller than 1 MSD.

Mistake

Student adds RELATIVE errors when computing the error of a sum or difference, or adds ABSOLUTE errors when computing the error of a product or quotient.

Root cause: formula misuse

Correction

Use Delta_Z = Delta_A + Delta_B for sums/differences (absolute errors add). Use Delta_Z/Z = Delta_A/A + Delta_B/B for products/quotients (relative errors add). They are NOT interchangeable — the rule is dictated by whether the operation is additive or multiplicative.

Wrong option pattern

Distractor option uses the wrong rule (e.g. quotes a small relative error for a sum where absolute errors should add).

Mistake

Student applies the 'fewest significant figures' rule for addition/subtraction (which actually uses fewest decimal places).

Root cause: concept gap

Correction

For multiplication/division, the result has the fewest significant figures of any input. For addition/subtraction, the result has the fewest decimal places of any input. Decimal places ≠ significant figures.

Wrong option pattern

Distractor truncates a sum to too few significant figures by applying the multiplication rule.

Formulas in this unit

Error in a power expression

Z = A^{p} B^{q} C^{r} \Rightarrow \frac{Δ Z}{Z} = ∣ p ∣ \frac{Δ A}{A} + ∣ q ∣ \frac{Δ B}{B} + ∣ r ∣ \frac{Δ C}{C}

The maximum relative error in a power expression is the sum of the absolute exponents weighted by the relative errors of the bases. Negative exponents (divisions) still take the |.| value because we want the worst-case error.

Symbol	Quantity	SI Unit
Z	Result	(combined)
p, q, r	Exponents (signed)	(dimensionless)
A, B, C	Measured quantities	(measured)

Valid when

Use absolute values of exponents — signs do not cancel in worst-case error analysis
Independent measurements assumption

Combination of errors — product or quotient

Z = A B or \frac{A}{B} \Rightarrow \frac{Δ Z}{Z} = \frac{Δ A}{A} + \frac{Δ B}{B}

When two measured quantities are multiplied or divided, the maximum RELATIVE errors add. The absolute error in the result is then Delta_Z = Z * (relative-error sum).

Symbol	Quantity	SI Unit
Z	Result of product/quotient	(combined unit)
A	First measured quantity	(measured)
B	Second measured quantity	(measured)
Delta_A/A	Relative error in A	(dimensionless)
Delta_B/B	Relative error in B	(dimensionless)

Valid when

A and B are independent measurements
Errors are quoted as maximum absolute uncertainties (worst-case)
For powers (Z = A^p * B^q), the rule generalises: Delta_Z/Z = |p|*Delta_A/A + |q|*Delta_B/B

Do NOT use when

Quantities are added or subtracted (use absolute-error rule instead)

Combination of errors — sum or difference

Z = A \pm B \Rightarrow Δ Z = Δ A + Δ B

When two quantities are added or subtracted, the maximum absolute errors of the inputs simply add to give the maximum absolute error of the output. The relative error is NOT what adds in this case.

Symbol	Quantity	SI Unit
Z	Result of sum/difference	(same as A,B)
A	First measured quantity	(measured)
B	Second measured quantity	(measured)
Delta_Z	Maximum absolute error in Z	(same as A,B)
Delta_A	Maximum absolute error in A	(same as A)
Delta_B	Maximum absolute error in B	(same as B)

Valid when

A and B are independent measurements (no correlated errors)
Errors are quoted as maximum absolute uncertainties (not standard deviations)
Use this rule for ADDITION or SUBTRACTION only — NOT for product/quotient

Do NOT use when

Quantities are multiplied or divided (use relative-error rule instead)
Errors are statistical (standard deviations) — quadrature-sum rule applies

NEET question patterns in this unit

Given dimensions of base/derived quantities, deduce the dimensions of a combined or unfamiliar quantity. Or: given a dimensional formula (e.g. [M L T⁻² A⁻²]), identify which physical quantity it represents. Approach: write the SI dimensional formula of each candidate, match exactly. Common shape: ratio E/G, F = αt²+βt finding dimensionless factor, [MLT⁻²A⁻²] → permeability.

InterpretationMedium

Common distractors

miscounts power of T

Off-by-one on time exponent (e.g. -1 vs -2)

Both plane angle (radian) and solid angle (steradian) are DIMENSIONLESS quantities even though they have unit names. Distractors include 'has dimensions of length' / 'angular' etc. Common form: 'plane angle and solid angle have ___' (1) different dimensions (2) same dimensions (3) units but no dimensions (4) ...

RecallEasy

Common distractors

treats radian as dimensional

Confuses 'has unit' with 'has dimensions'

Density ρ = m/V where V depends on measured dimensions raised to powers (e.g. cylindrical wire V = πr²L). Apply combination of errors: Δρ/ρ = Δm/m + 2 Δr/r + ΔL/L (radius gets factor of 2 from r²). Common shape: wire with mass, radius, length each ± uncertainty; find max % error in density. Distractors test (i) forgetting the 2× on radius, (ii) using absolute instead of relative errors.

Multi StepMedium

Common distractors

forgets power of two on radius

Default to summing all relative errors with weight 1

A derived quantity P depends on multiple measured quantities raised to various powers (e.g. P = a³ b² / (c d)); given % errors in each measurement, find max % error in P. Apply ΔP/P = 3 Δa/a + 2 Δb/b + Δc/c + Δd/d (powers as multipliers). Distractors test missed power factors and missing 'd' in denominator.

Multi StepMedium

Common distractors

linear sum no powers

Adding all relative errors with weight 1

Classify a described error source into the standard taxonomy: {personal, instrumental, least_count, random, systematic}. Common shape: 'errors due to unpredictable fluctuations in temperature/voltage' → RANDOM. 'Zero error of vernier' → instrumental. 'Eye-position bias' → personal.

RecallEasy

Common distractors

instrumental misclassified as random

Both feel 'unavoidable'; need to recognise instrumental ≠ random

Multiplication of two measured quantities (e.g. length × breadth = area); the result must have the FEWEST significant figures of any input. Common shape: 55.3 m × 25 m = 1382.5; with 2 sig figs in 25, round to 1400 m². Distractors test (i) keeping all digits, (ii) using decimal-place rule.

Direct ApplicationEasy

Common distractors

no rounding

Calculator answer kept verbatim