An equation is dimensionally correct only if the dimensions of every term on both sides of the equation are the same. The principle of homogeneity is the basis of dimensional analysis.
-- NCERT Class 11 Physics, Ch. 1, p. 7Dimensional Analysis
8 MCQs3 revision cards9-step worked example
Source: NCERT Units and MeasurementsPYQ coverage: NEET 2020, 2021, 2022, 2023, 2024, 2025Official key: NTA-verifiedLast reviewed: May 2026
Lesson
Dimensional analysis: the time-exponent trap that costs you marks.
The most common way NEET loses you a mark on dimensional analysis is the off-by-one error on the time exponent. You write [M L T⁻¹] when the answer is [M L T⁻²], or vice versa. This happens because students manipulate dimensional formulas in their heads instead of writing each quantity's dimensions explicitly and combining step by step.
What dimensional analysis actually is. Every physical quantity can be expressed as a product of powers of the seven base quantities (M, L, T, A, K, mol, cd). Dimensional analysis checks whether an equation is dimensionally consistent — if both sides don't match, the equation is certainly wrong (NCERT Class 11 Physics Chapter 1, page 7). It can also derive relations between quantities when you know which variables are involved (NCERT Class 11 Physics Chapter 1, page 8).
Key facts for NEET.
- Dimensional formulas are written as [Mᵃ Lᵇ Tᶜ ...]. The exponents a, b, c are called "dimensions."
- A dimensionally correct equation may still be numerically wrong — dimensional analysis cannot catch pure numbers or dimensionless functions (sin, cos, exp).
- Plane angle (radian) and solid angle (steradian) are dimensionless. They have SI unit names but no dimensions — radian is arc length ÷ radius, steradian is surface area ÷ r². Treating them as dimensional because they have unit names is a documented NEET trap.
The time-exponent trap in practice. When a question asks for the dimensions of a combined quantity (e.g., E/G, or coefficients in F = αt² + βt), you must decompose every quantity into [M L T ...] and subtract exponents carefully. The T exponent is where off-by-one errors cluster: energy has T⁻², force has T⁻², velocity has T⁻¹. Mixing these up by one power is the single most reliable way to land on a distractor.
Counter-strategy: Write the full dimensional formula of every quantity before combining. Never shortcut exponent arithmetic.
Practice MCQs
Select an option to see the explanation. Wrong answers show why your choice was tempting — and name the exact trap it exploits.
MCQ 1Easy RecallPractice
Which of the following statements about dimensional analysis is correct?
MCQ 2Direct ApplicationPractice
The dimensional formula of energy is:
MCQ 3Easy RecallPYQ Pattern
Plane angle and solid angle are:
MCQ 4Direct ApplicationPractice
The force on a body is given by F = αt² + βt, where t is time. The dimensions of α are:
MCQ 5Direct ApplicationPractice
The dimensional formula [M L⁻¹ T⁻²] corresponds to:
MCQ 6Direct ApplicationPractice
If velocity (v), force (F), and time (T) are chosen as fundamental quantities, the dimensions of mass in this new system are:
MCQ 7Concept TrapPractice
A student claims that the equation v = u + at² is correct because both sides have the dimensions of velocity. Which of the following is the best response?
MCQ 8CalculationPYQ Pattern
The velocity v of a wave on a stretched string depends on the tension T (force) and the linear mass density μ (mass per unit length). Using dimensional analysis, the relation is v = k Tᵃ μᵇ. The values of a and b are:
Quick recall before you leave
Worked Example
Pattern: Dimensions of a derived quantity (NEET pattern: dimensions of derived quantity, frequency 4, highest among in-scope patterns)
- 1
Given
A physical quantity X is defined as X = Energy / (Gravitational constant), i.e. X = E / G. Dimensions of energy: [M L² T⁻²] Dimensions of gravitational constant G: [M⁻¹ L³ T⁻²]
- 2
Required
Find the dimensional formula of X.
- 3
Concept
Dimensional analysis: to find dimensions of a ratio, subtract the exponents of the denominator from those of the numerator for each base dimension.
- 4
Formula
[X] = [E] / [G] = [Mᵃ Lᵇ Tᶜ] where each exponent is found by subtracting.
- 5
Substitution
[X] = [M L² T⁻²] / [M⁻¹ L³ T⁻²] For M: 1 − (−1) = 2 For L: 2 − 3 = −1 For T: (−2) − (−2) = 0
- 6
Calculation
[X] = [M² L⁻¹ T⁰] = [M² L⁻¹] Note: No numerical constants appear in this problem. The calculation is purely exponent arithmetic.
- 7
Final answer
**X = E/G has dimensions [M² L⁻¹].**
- 8
Common trap
The time-exponent trap: students often get T⁰ wrong here because both E and G have T⁻². When you subtract (−2) − (−2), the result is 0 — the time dimension cancels completely. A common error is writing T⁻⁴ (adding the exponents instead of subtracting) or T⁻² (forgetting to subtract the denominator's T exponent at all). Always write out the subtraction explicitly: numerator exponent MINUS denominator exponent.
- 9
Similar NEET-style question
"The dimensions of E²/G (where E is energy and G is gravitational constant) are:" Approach: [E²/G] = [M L² T⁻²]² / [M⁻¹ L³ T⁻²] = [M² L⁴ T⁻⁴] / [M⁻¹ L³ T⁻²] = [M³ L¹ T⁻²]. Track each exponent separately. The T exponent is (−4) − (−2) = −2, not −4 or 0. ---
Before solving, remember these
Dimensional analysis is used to: (i) check the dimensional correctness of an equation, (ii) convert a physical quantity from one system of units to another, and (iii) deduce a relation among physical quantities (subject to the limitation that dimensionless constants cannot be obtained).
-- NCERT Class 11 Physics, Ch. 1, p. 8Formulas
3 formulas — click to collapse
Error in a power expression
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
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
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
Exam Traps & Common Mistakes
These are the exact patterns that cause wrong answers in NEET. Each trap includes when it triggers and how to avoid it.
9 items — click to collapse
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.
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.
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).
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).
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.
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.
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.
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).
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.
Past Year Questions
13 questions from NEET 2020, 2021, 2022, 2023, 2024, 2025. Answers verified against NTA official keys. — click to collapse
NEET 2025
15.00 cm
25.18 cm
35.08 cm
44.98 cm
NTA Answer: Option 4(final)
NEET 2025
115%
210%
32%
413%
NTA Answer: Option 4(final)
NEET 2024Revised key
The quantities which have the same dimensions as those of solid angle are :
1strain and angle
2stress and angle
3strain and arc
4angular speed and stress
NTA Answer: Option 1(revised_final)
NEET 2024Revised key
1α
2αt β
3αβt αβ
4t
NTA Answer: Option 2(revised_final)
NEET 2023
1Personal errors
2Least count errors
3Random errors
4Instrumental errors
NTA Answer: Option 3(final)
NEET 2023
11.3%
21.6%
31.4%
41.2%
NTA Answer: Option 2(final)
NEET 2022
The dimensions [MLT–2A–2] belong to the
1Electric permittivity
2Magnetic flux
3Self inductance
4Magnetic permeability
NTA Answer: Option 4(final)
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)
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)
NEET 2022
114 102
2138 101
31382
41382.5
NTA Answer: Option 1(final)
NEET 2021
10.052 cm
20.52 cm
30.026 cm
40.26 cm
NTA Answer: Option 1(final)
NEET 2020
19.9 m
29.9801 m
39.98 m
49.980 m Hindi+English 9 H3
NTA Answer: Option 3(final)
NEET 2020
11.0 mm
20.01 mm
30.25 mm
40.5 mm Hindi+English 11 H3
NTA Answer: Option 4(final)
How NEET usually asks this
9 recurring patterns from past papers — click to collapse
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)
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
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'
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
Given the least count and the number of circular-scale divisions of a screw gauge, find the pitch (or vice versa). Formula: least count = pitch / (number of circular scale divisions). Common shape: LC = 0.01 mm, 50 divisions; find pitch (= 0.5 mm).
Direct ApplicationEasy
Common distractors
swaps pitch and LC
Confusing which side of the formula is asked
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
Subtraction of two measured quantities with very different decimal places; the answer must reflect the FEWEST decimal places (not the fewest significant figures). Common shape: 9.99 m - 0.0099 m, or similar. Distractors test (i) using sig-fig rule from multiplication/division, (ii) keeping all digits unchanged, (iii) over-rounding to 1-2 sig figs.
Direct ApplicationEasy
Common distractors
applies mult rule to subtraction
Default to 'fewest sig figs' without distinguishing subtraction's decimal-places rule
Vernier calipers: (N+1) divisions of vernier scale coincide with N divisions of main scale; given main-scale division (1 MSD), find vernier constant (least count). Formula: VC = 1 MSD - 1 VSD = 1 MSD × (1 - N/(N+1)) = 1 MSD / (N+1). Common shape: 1 MSD = 0.1 mm, k VSD = (k+1) MSD; find VC.
Direct ApplicationEasy
Common distractors
swap N and N+1
Confusing which side has the larger count
Sources
NCERT refs: Class 11 Physics Chapter 1, p.7 | Class 11 Physics Chapter 1, p.8
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