Units & Measurements— Notes, Formulas & Revision

SI base units: m (length), kg (mass), s (time), A (current), K (temp), mol (amount), cd (luminosity).

CGS: cm, g, s. 1 N = 10⁵ dyne; 1 J = 10⁷ erg.

Always multiply by 1 in disguise: e.g., (1 m / 100 cm) to cancel units.

Derived units: N = kg·m/s², J = N·m, Pa = N/m², W = J/s.

All non-zero digits are significant; zeros between non-zero digits are significant.

Leading zeros are NOT significant; trailing zeros after a decimal ARE.

In multiplication/division, result has as many sig figs as the LEAST-precise factor.

In addition/subtraction, result has as few DECIMAL PLACES as the least-precise term.

Rounding introduces a small error each time — errors accumulate over many operations.

Carry one extra digit through intermediate calculations; round only the final answer.

For n additions with error δ each, total error grows as √n (random) or n (systematic).

Absolute error: Δa = |a_measured − a_true|.

Relative error: Δa/a. Percentage error: (Δa/a)×100%.

Mean absolute error: average of |aᵢ − a_mean| over n readings.

For R = A ± B: ΔR = ΔA + ΔB (absolute errors add).

The relative error in a sum is NOT simply sum of relative errors — use absolute.

For R = AB or A/B: ΔR/R = ΔA/A + ΔB/B (relative errors add).

For R = Aⁿ: ΔR/R = n·ΔA/A.

Dimensional formula: [quantity] = MᵃLᵇTᶜ. E.g., [F] = MLT⁻², [E] = ML²T⁻².

Dimensionless quantities: angle (rad), refractive index, strain, etc.

Two quantities with the same dimensions can be added or equated.

A valid physical equation must be dimensionally homogeneous — all terms share the same dimensions.

Dimensional check is necessary but not sufficient — it can catch wrong equations but not confirm correct ones.

Assume quantity Q depends on parameters xᵢ through powers: Q = k·x₁ᵃ·x₂ᵇ·…

Match dimensions on both sides to solve for a, b, … .

Final answer has an undetermined dimensionless constant k (often 1, 2π, etc.)

Least count (LC) = value of 1 main-scale division − value of 1 vernier-scale division.

Typical LC = 0.1 mm = 0.01 cm for a 10-division vernier.

Reading = main-scale reading (MSR) + (vernier-scale division that aligns) × LC.

Check for zero error first; subtract it from all readings.

Least count = pitch / number of circular divisions (typically 0.5 mm / 50 = 0.01 mm).

Reading = linear-scale reading + (circular-scale reading) × LC.

Zero error must be applied: corrected reading = observed − zero error.

Zero error: reading when the instrument shows zero but the actual is nonzero.

Positive zero error: instrument reads above zero → SUBTRACT from measurements.

Negative zero error: reads below zero → ADD to measurements.

Corrected reading = observed − zero error.