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⚛️ Physics · Class 11 · NEET & JEE

Units and Measurements

SI base units, dimensional analysis, significant figures, and error propagation. The toolkit every physics calculation relies on.

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Reading time~11 min

Revision time~4 min

Last updated2026-08-18

1 Read the chapter ~11 min

🎯 Key Points

SI system has 7 base units: metre (length), kilogram (mass), second (time), ampere (electric current), kelvin (temperature), mole (amount of substance), candela (luminous intensity)
Dimensional formula expresses a quantity in powers of [M L T ...]; the principle of homogeneity says both sides of a correct physical equation must have the same dimensions
Significant figures reflect measurement precision: in multiplication/division, the result keeps as many significant figures as the LEAST precise measurement; in addition/subtraction, the result keeps as many decimal places as the measurement with the FEWEST decimal places
Absolute error = |measured value − true/mean value|; Relative error = mean absolute error ÷ mean value; Percentage error = relative error × 100
Accuracy = closeness to the true value; Precision = reproducibility among repeated measurements - a result can be precise (consistent) without being accurate (correct)

The 7 SI base units. All derived units (newton, joule, watt, volt, etc.) can be expressed as combinations of these 7 fundamental units raised to various powers.

The SI System of Units

Length (metre, m), Mass (kilogram, kg), Time (second, s), Electric current (ampere, A), Thermodynamic temperature (kelvin, K), Amount of substance (mole, mol), Luminous intensity (candela, cd)
Two dimensionless units used for angles: radian (rad) for plane angle, steradian (sr) for solid angle - these have no dimensional formula since they are ratios of lengths
Every other physical quantity (velocity, force, energy, etc.) is a derived unit, built by combining powers of the 7 base units, e.g. velocity = m/s, force (newton) = kg·m/s²

Dimensional Analysis

Dimensional formula: expresses a derived quantity as [M^a L^b T^c ...], the powers to which base quantities (mass, length, time, etc.) must be raised; e.g. force = [M L T⁻²], energy = [M L² T⁻²]
Principle of homogeneity of dimensions: a physical equation is dimensionally correct only if both sides have the identical dimensional formula; this lets you check an equation's correctness without knowing the actual values
Uses: (1) checking the dimensional consistency of an equation, (2) converting a physical quantity from one system of units to another, (3) deriving a relationship between physical quantities (up to an unknown dimensionless constant)
Limitations: cannot determine dimensionless proportionality constants (like 2π or ½); cannot be applied to equations involving trigonometric, exponential, or logarithmic functions; cannot distinguish between two quantities that happen to share the same dimensions (e.g. work and torque are both [M L² T⁻²])

Significant Figures

All non-zero digits are significant; zeros between two non-zero digits are significant (e.g. 1005 has 4 significant figures)
Leading zeros (before the first non-zero digit) are never significant - they only locate the decimal point, e.g. 0.0025 has just 2 significant figures
Trailing zeros after a decimal point ARE significant, e.g. 2.500 has 4 significant figures; trailing zeros in a number with no decimal point are ambiguous, which is why scientific notation (e.g. 2.5 × 10³) is preferred for clarity
Rounding off rule: if the digit to be dropped is more than 5, round up; if less than 5, round down; if exactly 5, round to the nearest even digit

Errors in Measurement

Systematic errors: consistent, repeatable errors from instrumental flaws, imperfect experimental technique, or personal bias - these can be reduced/corrected by improving technique or calibration
Random errors: irregular fluctuations with no fixed pattern, arising from unpredictable variations during measurement - reduced by taking many readings and averaging
Absolute error (Δa) = |individual measured value − mean value|; Mean absolute error = average of all individual absolute errors
Relative (fractional) error = mean absolute error ÷ mean value; Percentage error = relative error × 100
Combination of errors: for sums/differences, absolute errors add directly; for products/quotients, relative errors add; for a quantity raised to a power p, the relative error gets multiplied by p

Error Propagation (Combination of Errors)

Sum or difference: if Z = A + B or Z = A − B, the absolute errors ADD: ΔZ = ΔA + ΔB. Note the error adds even for a difference - you take the worst case, never subtract errors.
Product or quotient: if Z = AB or Z = A/B, the relative (fractional) errors ADD: ΔZ/Z = ΔA/A + ΔB/B.
Power: if Z = Aᵖ Bᑫ / Cʳ, then ΔZ/Z = p(ΔA/A) + q(ΔB/B) + r(ΔC/C). Every term is positive; a quantity raised to a higher power contributes proportionally more error.
Rule of thumb: measure the quantity that enters with the highest power most carefully, since its error is magnified the most in the final result.

Least Count and Measuring Instruments

Least count (LC) = smallest measurement an instrument can read reliably; it sets the resolution and the limit of instrumental error for a single reading.
Vernier calliper: LC = 1 main scale division − 1 vernier scale division = value of 1 MSD ÷ number of vernier divisions. A standard vernier has LC = 0.1 mm (0.01 cm).
Screw gauge / micrometer: LC = pitch ÷ number of circular (head) scale divisions, where pitch = distance moved per full rotation of the screw. A typical screw gauge has LC = 0.01 mm.
Zero error: if the instrument does not read zero when it should, subtract this zero error (with sign) from every reading. Positive zero error is subtracted; negative zero error is added back.
Reading = main scale reading + (vernier/circular scale coincidence × least count), corrected for zero error.

Accuracy, Precision, and Number of Significant Figures

Accuracy is how close a reading is to the true value; precision is how finely and consistently the instrument reads (governed by its least count). High precision does not guarantee accuracy - a mis-calibrated instrument gives precise but inaccurate values.
The number of significant figures in a result reflects its precision: writing 2.30 m implies a precision of ±0.01 m, whereas 2.3 m implies ±0.1 m.
Order of magnitude: the power of ten nearest to a quantity (e.g. 2.7 × 10³ has order of magnitude 10³). Useful for quick estimation and checking whether an answer is reasonable.

🚀 JEE Advanced Edge

Dimensional analysis finds the FORM, not the exact equation: Using dimensions, you can show that a simple pendulum's time period must follow T ∝ √(l/g), but dimensional analysis can never tell you the proportionality constant is 2π - that factor only emerges from the actual differential-equation derivation. JEE frequently sets up "derive the dependence" questions that test exactly this boundary between what dimensional analysis can and cannot give you.

The power rule is the most commonly mis-applied error-propagation rule: For Z = A^pB^q/C^r, the relative error is ΔZ/Z = p(ΔA/A) + q(ΔB/B) + r(ΔC/C) - note that errors in the DENOMINATOR's quantities still ADD (they never subtract), because you're combining worst-case uncertainty, not a derivative with sign. Many students wrongly subtract the denominator's relative error term.

Worked problem: The radius of a sphere is measured with a 1% relative error. What is the percentage error in its calculated volume? Since V = (4/3)πr³, V depends on r raised to the power 3, so by the power rule, the relative error in V is 3 × (relative error in r) = 3 × 1% = 3%. This is why measuring a length precisely matters far more when it is cubed in a later calculation than when it appears only to the first power.

Fundamental and Derived Quantities

Fundamental (base) quantities are independent and cannot be expressed in terms of others; SI has seven: length, mass, time, electric current, temperature, amount of substance, luminous intensity.
Derived quantities are obtained by combining base quantities, e.g. speed = length/time, force = mass × acceleration.
The seven SI base units are metre (m), kilogram (kg), second (s), ampere (A), kelvin (K), mole (mol) and candela (cd).
Supplementary units: radian (rad) for plane angle and steradian (sr) for solid angle.
A complete set of base units together with derived units forms a system of units (SI, CGS, FPS, MKS).

Measurement of Large Distances: Parallax Method

Parallax is the apparent shift in position of an object against a distant background when viewed from two different points.
If the basis (distance between the two observation points) is b and the parallax angle is θ, the distance is D = b/θ, with θ in radians.
Used to find distances of nearby stars and planets that are too large to measure directly.
The angular diameter of a planet is measured, and its actual diameter d = α × D, where α is the angular size in radians.
Astronomical distances use units like the astronomical unit (AU), light year and parsec (1 parsec is about 3.08 × 10¹⁶ m).

Order of Magnitude and Scientific Notation

A number is written in scientific notation as a × 10^b, where a lies between 1 and 10 and b is an integer.
The order of magnitude is the power of ten nearest to the number; if a is greater than 5, round the power up by one.
Example: 4500 m is 4.5 × 10³ m, so its order of magnitude is 3.
Order-of-magnitude estimates give a quick check on whether an answer is physically reasonable.
Ranges of physical quantities span enormous orders of magnitude, from the size of a proton (about 10^-15 m) to the size of the observable universe (about 10²⁶ m).

Applications and Limitations of Dimensional Analysis

Applications: checking the dimensional correctness of an equation, converting a quantity from one system of units to another, and deriving relations among physical quantities.
Unit conversion uses n₁[u₁] = n₂[u₂]; the numerical value is inversely proportional to the size of the unit.
Limitation: it cannot determine dimensionless constants (like 1/2 or 2π) in a formula.
It cannot derive relations involving trigonometric, exponential or logarithmic functions.
It fails when a quantity depends on more than three physical factors, or when a sum of terms with the same dimensions is involved.

2 Revise ~4 min before the exam

📐 Formula Sheet

Seven SI base units: metre (length), kilogram (mass), second (time), ampere (current), kelvin (temperature), mole (amount), candela (luminous intensity)
Dimensional formulae: force [MLT⁻²], energy/work [ML²T⁻²], power [ML²T⁻³], pressure [ML⁻¹T⁻²], momentum [MLT⁻¹]
Principle of homogeneity: every term in a valid equation has the same dimensions
Absolute error: Δa = |a_mean − a_i| | Relative error: Δa/a | Percentage error: (Δa/a) × 100
Sums/differences: absolute errors add | Products/quotients: relative errors add
Powers: if Z = Aᵖ, then ΔZ/Z = p·(ΔA/A)
Significant figures: in × and ÷, the result carries the fewest sig figs of the inputs; in + and −, the fewest decimal places

3 Practice apply it

✍️ Worked Examples

Example 1 - Checking an equation dimensionally
Q: Verify whether v² = u² + 2as is dimensionally correct.
Step 1 - Left side: [v²] = [LT⁻¹]² = [L²T⁻²].
Step 2 - First term on the right: [u²] = [L²T⁻²]. ✓
Step 3 - Second term: [as] = [LT⁻²][L] = [L²T⁻²]. ✓
Answer: all three terms are [L²T⁻²], so the equation is dimensionally consistent. Caution: dimensional analysis cannot detect a wrong numerical factor - it would equally "accept" v² = u² + 5as.

Example 2 - Error propagation in a product
Q: A rectangle measures 5.0 ± 0.1 cm by 4.0 ± 0.1 cm. Find its area and the percentage error.
Step 1 - Area: A = 5.0 × 4.0 = 20 cm².
Step 2 - For a product, relative errors add: ΔA/A = Δl/l + Δb/b = 0.1/5.0 + 0.1/4.0 = 0.02 + 0.025 = 0.045.
Step 3 - Percentage: 0.045 × 100 = 4.5%.
Answer: A = 20 cm² ± 4.5% (i.e. ± 0.9 cm²). Trap: absolute errors add for sums, not products - don't write ΔA = 0.2.

Example 3 - Error in a power
Q: The period of a pendulum is measured with 2% error. What is the resulting percentage error in g, given T = 2π√(L/g)?
Step 1 - Rearrange: squaring gives T² = 4π²L/g, so g = 4π²L/T².
Step 2 - g depends on T⁻², so Δg/g = 2 × (ΔT/T) (the exponent multiplies the error).
Step 3 - Substitute: 2 × 2% = 4%.
Answer: 4%. Note: this is why timing many oscillations at once, rather than one, matters so much in the lab.