Units and Measurements formulas for JEE Main & Advanced

23 formula entries for Units and Measurements, in the order the chapter builds them. Each shows the formula, when to use it and the first trap to avoid.

Physical Quantity and Unit

Physical Quantity = n \times u

When to use. Whenever converting between units or expressing measurements.

Trap. Confusing fundamental and derived quantities (e.g., area is derived, not fundamental).

Fundamental and Derived Units

Derived unit = [M^{a} L^{b} T^{c} \dots]

When to use. When analyzing dimensional formulas or checking consistency of equations.

Trap. Mistaking radian or steradian as fundamental (they are supplementary).

SI Prefixes

1 km = 10^{3} m

1 mm = 10^{- 3} m

1 μ m = 10^{- 6} m

1 nm = 10^{- 9} m

When to use. When converting between metric units or expressing very large/small quantities.

Trap. Confusing micro ( $μ$ ) with milli (m) or nano (n).

Dimensional Formula

[Velocity] = [M^{0} L^{1} T^{- 1}]

When to use. To check equation consistency, derive relations, or convert units.

Trap. Forgetting that dimensionless constants (like $π$ , $e$ ) have no dimensions.

Dimensional Analysis for Equation Consistency

[L H S] = [R H S]

When to use. When verifying derived formulas or checking if an equation is plausible.

Trap. Assuming dimensional correctness implies physical correctness (it only checks consistency).

Deriving Relations Using Dimensional Analysis

T = k l^{a} g^{b}

→

[T] = [L]^{a} [L T^{- 2}]^{b}

→

a = 1 / 2, b = - 1 / 2

When to use. When you need to guess the formula for a new quantity based on known dependencies.

Trap. Forgetting that dimensionless constants (like $2 π$ ) cannot be determined by dimensions.

Significant Figures

Example:

0.00450

has 3 significant figures (4,5,0).

When to use. When reporting measurements or performing calculations with limited precision.

Trap. Counting leading zeros as significant (e.g., $0.002$ has 1, not 3).

Arithmetic with Significant Figures

Example:

12.11 + 0.3 = 12.4

(one decimal).

2.5 \times 3.42 = 8.6

(two sig figs).

When to use. When performing calculations with measured values to avoid false precision.

Trap. Rounding incorrectly (e.g., $2.5 \times 3.42 = 8.55$ rounded to 8.6, not 8.5).

Accuracy and Precision

No formula, but concept: accurate = small systematic error; precise = small random error.

When to use. When evaluating experimental data or instrument quality.

Trap. Using the terms interchangeably (they are distinct).

Error in Measurement

Relative error = \frac{Δ a}{a_{mean}}

Percentage error = \frac{Δ a}{a_{mean}} \times 100 %

When to use. When analyzing experimental data or estimating uncertainty.

Trap. Confusing absolute error with relative error.

Propagation of Errors (Addition/Subtraction)

Δ Z = Δ A + Δ B

When to use. When combining measurements through addition or subtraction.

Trap. Using relative error instead of absolute error for addition/subtraction.

Propagation of Errors (Multiplication/Division)

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

When to use. When combining measurements through multiplication or division.

Trap. Using absolute errors instead of relative errors.

Propagation of Errors (Powers and Functions)

\frac{Δ Z}{Z} = n \frac{Δ A}{A}

(for

Z = A^{n}

);

Δ Z = ∣ \frac{d Z}{d A} ∣ Δ A

(general)

When to use. When dealing with powers, roots, or trigonometric functions of measured quantities.

Trap. Forgetting to convert angles to radians for trigonometric functions.

Least Count and Instrument Error

Least count = \frac{value of 1 main scale division}{number of vernier divisions}

(for vernier)

When to use. When reading measurements from instruments or calculating instrument precision.

Trap. Confusing least count with accuracy (least count is a limit, not the actual error).

Vernier Caliper Reading

Reading = MSR + (VCD \times LC)

;

Corrected reading = Reading - Zero error

When to use. When measuring length with a vernier caliper.

Trap. Forgetting to check for zero error (positive or negative).

Screw Gauge Reading

Reading = MSR + (CSR \times LC)

;

LC = \frac{pitch}{number of circular divisions}

When to use. When measuring small thicknesses (e.g., wire diameter).

Trap. Forgetting that pitch is the distance moved per rotation (usually 0.5 mm or 1 mm).

Order of Magnitude

N = a \times 10^{b}

with

1 \leq a < 10

; order

= 10^{b}

a \leq 5

, else

10^{b + 1}

(NCERT convention)

When to use. When estimating or comparing very large/small quantities.

Trap. Order of magnitude is the power of 10 nearest the number: with the NCERT cutoff, if $a \leq 5$ the order is $10^{b}$ , otherwise $10^{b + 1}$ (e.g. $8 \times 10^{4}$ has order $10^{5}$ ).

Common Physical Constants and Their Dimensions

[h] = [M L^{2} T^{- 1}]

[G] = [M^{- 1} L^{3} T^{- 2}]

[k] = [M L^{2} T^{- 2} K^{- 1}]

[ϵ_{0}] = [M^{- 1} L^{- 3} T^{4} A^{2}]

When to use. When using dimensional analysis or deriving formulas involving these constants.

Trap. Forgetting that $G$ has negative mass dimension.

Conversion of Units Using Dimensions

n_{2} = n_{1} {(\frac{M_{1}}{M_{2}})}^{a} {(\frac{L_{1}}{L_{2}})}^{b} {(\frac{T_{1}}{T_{2}})}^{c}

When to use. When converting between different unit systems (e.g., CGS to SI).

Trap. Forgetting to raise the ratios to the appropriate powers (e.g., for force, $a = 1, b = 1, c = - 2$ ).

Parallax Method for Distance Measurement

θ = \frac{b}{D}

(small angle approximation);

D = \frac{b}{θ}

When to use. When estimating astronomical distances (e.g., distance to a star).

Trap. Forgetting to convert parallax angle to radians (not arcseconds).

Range of Lengths and Orders of Magnitude

No formula; memorize approximate values.

When to use. When estimating or comparing scales in physics problems.

Trap. Mixing up atomic and nuclear sizes (nucleus is $10^{5}$ times smaller).

Range of Masses and Orders of Magnitude

No formula; memorize approximate values.

When to use. When estimating masses in problems (e.g., number of atoms in a sample).

Trap. Confusing mass of electron with proton (proton is ~1800 times heavier).

Range of Time Intervals and Orders of Magnitude

No formula; memorize approximate values.

When to use. When estimating time scales in physics (e.g., period of oscillation).

Trap. Confusing microsecond ( $10^{- 6}$ s) with nanosecond ( $10^{- 9}$ s).

Values to remember

$10^{3}$ SI Prefixes
$10^{- 6}$ SI Prefixes
$10^{- 9}$ SI Prefixes
$h = 6.63 \times 10^{- 34} J s$ Common Physical Constants and Their Dimensions
$G = 6.67 \times 10^{- 11} {N m}^{2} / {kg}^{2}$ Common Physical Constants and Their Dimensions
$1 parsec = 3.26 ly = 3.08 \times 10^{16} m$ Parallax Method for Distance Measurement
$1 \overset{˚}{A} = 10^{- 10} m$ Range of Lengths and Orders of Magnitude
$1 fm = 10^{- 15} m$ Range of Lengths and Orders of Magnitude
$m_{e} = 9.1 \times 10^{- 31} kg$ Range of Masses and Orders of Magnitude
$m_{p} = 1.67 \times 10^{- 27} kg$ Range of Masses and Orders of Magnitude
$1 year = 3.15 \times 10^{7} s$ Range of Time Intervals and Orders of Magnitude
$1 day = 86400 s$ Range of Time Intervals and Orders of Magnitude

Read the full chapter: Units and Measurements for JEE, with what JEE asks and the common traps.