The trap that costs you marks here: getting the time exponent wrong by one. A student writes the dimensions of energy as [M L² T⁻¹] instead of [M L² T⁻²], or writes force as [M L T⁻¹] instead of [M L T⁻²]. One wrong exponent, one lost mark. NEET has tested this pattern repeatedly from 2021 through 2024.
What dimensions actually are. Every physical quantity can be expressed as a product of powers of the seven SI base quantities: mass (M), length (L), time (T), electric current (A), thermodynamic temperature (K), amount of substance (mol), and luminous intensity (cd). The dimensional formula of a quantity is this power-product written in square brackets. For example, force = mass × acceleration = M × L T⁻² = [M L T⁻²]. This is defined in NCERT Class 11 Physics Chapter 1, page 7.
The radian/steradian confusion. Plane angle (radian) and solid angle (steradian) are ratios — arc length divided by radius, or surface area divided by radius squared. They are dimensionless: [M⁰ L⁰ T⁰]. The fact that they have named units does not give them dimensions. NEET 2022 tested exactly this distinction.
How NEET tests dimensions. The common pattern: you are given a derived or unfamiliar combination of quantities and asked to find its dimensional formula, or you are given a dimensional formula like [M L T⁻² A⁻²] and asked which physical quantity it represents. The approach is mechanical — write the dimensional formula of each constituent, combine using algebra, and match. The error that loses marks is sloppy exponent arithmetic, especially on the time dimension.
Watch-out. When a formula has multiple terms added together (like F = αt² + βt), each term must have the same dimensions as F. Use this dimensional homogeneity to find the dimensions of the unknown constants α and β. Do not assume — derive.
Select an option to see the explanation. Wrong answers show why your choice was tempting — and name the exact trap it exploits.
Which of the following is the correct dimensional formula for energy?
The dimensional formula [M L T⁻² A⁻²] represents which physical quantity?
Plane angle and solid angle are:
If force F is given by F = αt² + βt, where t is time and F has dimensions [M L T⁻²], what are the dimensions of α?
The dimensions of the ratio (energy / gravitational constant) are:
If velocity [v], force [F], and time [T] are chosen as fundamental quantities, the dimensional formula of mass in this new system is:
Which of the following quantities is dimensionless?
In the equation F = αt² + βt, where F is force [M L T⁻²] and t is time, the dimensions of β are:
Pattern: Deduce the dimensions of a derived combination of quantities (NEET 2022, 2024 pattern — dimensions of a derived quantity).
Given
A physical quantity P is defined as P = E²/(G · h), where: - E is energy - G is the universal gravitational constant - h is Planck's constant Find the dimensions of P.
Required
The dimensional formula of P = E²/(G · h).
Concept
Write the dimensional formula of each constituent quantity, then combine using the rules of exponents. The principle of dimensional homogeneity guarantees a unique result.
Formula
- E (energy) = [M L² T⁻²] - G (gravitational constant): from F = Gm₁m₂/r², G = [M⁻¹ L³ T⁻²] - h (Planck's constant): from E = hν, h = E/ν = [M L² T⁻²]/[T⁻¹] = [M L² T⁻¹]
Substitution
P = E² / (G · h) = [M L² T⁻²]² / ([M⁻¹ L³ T⁻²] · [M L² T⁻¹]) Numerator: [M² L⁴ T⁻⁴] Denominator: [M⁻¹ L³ T⁻²] · [M L² T⁻¹] = [M⁻¹⁺¹ L³⁺² T⁻²⁻¹] = [M⁰ L⁵ T⁻³]
Calculation
P = [M² L⁴ T⁻⁴] / [M⁰ L⁵ T⁻³] = [M²⁻⁰ L⁴⁻⁵ T⁻⁴⁻(⁻³)] = [M² L⁻¹ T⁻¹] Cross-check the time exponent carefully: −4 − (−3) = −4 + 3 = −1. This is the step where the off-by-one trap strikes — students who rush write T⁻² or T⁰. Note: the exponents 2 (on E) and 1 (on G and h) are exact mathematical powers defined by the formula P = E²/(Gh). They are not measurements and do not contribute to any error or significant-figure analysis.
Final answer
**P = [M² L⁻¹ T⁻¹]**
Common trap
The time-exponent off-by-one error. In the denominator, T⁻² × T⁻¹ = T⁻³ (not T⁻²). Then −4 − (−3) = −1 (not −2). Students who skip explicit exponent arithmetic and guess get T⁻² in the final answer, which is wrong.
Similar NEET-style question
Find the dimensions of the quantity σ = F²/(E · v), where F is force, E is energy, and v is velocity. (Answer: [M L⁻¹ T⁻²] — verify by writing each dimensional formula and combining.) ---
The dimensions of a physical quantity are the powers (exponents) to which the base quantities are raised to represent that quantity. The bracket [Q] denotes the dimensions of Q. For example, [velocity] = [L T⁻¹]; [force] = [M L T⁻²].
-- NCERT Class 11 Physics, Ch. 1, p. 7The 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) |
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) |
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) |
These are the exact patterns that cause wrong answers in NEET. Each trap includes when it triggers and how to avoid it.
Category: Similar Terms
Student gets the time exponent wrong by 1 (e.g. T⁻¹ vs T⁻²) when manipulating dimensional formulas.
Question asks for dimensions of a derived combination (e.g. E/G, F = αt² + βt) where time exponent matters.
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.
Question gives a derived quantity formula with mixed-power dependencies; asks for the max relative error. Distractors omit the power factor.
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.
Question describes an error source and asks for its taxonomic category. Distractors include cognate categories.
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.
Question gives one of (pitch, N, least count) and asks for another; distractors offer the inverted ratio or off-by-factor-of-10.
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.
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.
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.
Question asks about dimensions of plane angle, solid angle, or comparison.
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.
Question gives '(N+1) divisions of vernier coincide with N divisions of main' or similar phrasing.
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
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.
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
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.
Distractor truncates a sum to too few significant figures by applying the multiplication rule.
The quantities which have the same dimensions as those of solid angle are :
The dimensions [MLT–2A–2] belong to the
Plane angle and solid angle have
miscounts power of T
Off-by-one on time exponent (e.g. -1 vs -2)
forgets power of two on radius
Default to summing all relative errors with weight 1
linear sum no powers
Adding all relative errors with weight 1
treats radian as dimensional
Confuses 'has unit' with 'has dimensions'
instrumental misclassified as random
Both feel 'unavoidable'; need to recognise instrumental ≠ random
swaps pitch and LC
Confusing which side of the formula is asked
no rounding
Calculator answer kept verbatim
applies mult rule to subtraction
Default to 'fewest sig figs' without distinguishing subtraction's decimal-places rule
swap N and N+1
Confusing which side has the larger count
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