Units and Measurement — Exam Practice

Answer Significant figures communicate exactly how precisely a quantity was actually measured. Reporting too many digits falsely implies greater precision than the instrument achieved; reporting too few discards real, measured information.

Answer [s] = [L]. [ut] = [LT⁻¹][T] = [L]. [at²] = [LT⁻²][T²] = [L].

Every term has dimension [L], matching [s] — the equation is dimensionally consistent.

Principle of homogeneity: every term on both sides of a valid physical equation must carry identical dimensions. This gives a quick way to check whether a derived formula could possibly be correct (though matching dimensions alone doesn't guarantee full correctness — it can't catch a missing dimensionless constant).

Answer (a) Significant figures are the digits in a measurement that carry real precision. Rules: all non-zero digits are significant; zeros between non-zero digits are significant; leading zeros are never significant; trailing zeros after a decimal point are significant; trailing zeros in a whole number without a decimal point are ambiguous.

(b) Area = 15.2 × 8.3 = 126.16 cm² → rounded to 2 significant figures (matching the least-precise measurement, 8.3) = 1.3×10² cm².
% error = (Δl/l + Δb/b) × 100 = (0.1/15.2 + 0.1/8.3) × 100 ≈ (0.0066 + 0.0120) × 100 ≈ 1.9%.

Solution All non-zero digits (6, 3, 2) are significant, and the trailing zero after the decimal point is also significant — giving 4 total.

Solution Pressure = Force/Area = [MLT⁻²]/[L²] = [ML⁻¹T⁻²].

Solution Work = Force × distance = [ML²T⁻²]. Torque = Force × distance (perpendicular component) = [ML²T⁻²] — identical dimensions, even though they're conceptually different quantities (torque is a vector, work is a scalar).

Solution By measuring the small angular shift (parallax) of a distant object viewed from two separated points, D = b/θ gives the distance — impractical to measure directly by any other means at astronomical scales.

Solution [L/g] = [L] / [LT⁻²] = [T²]. Taking the square root: [T² ]^(1/2) = [T], matching [T] on the left-hand side.

The equation is dimensionally consistent (though this alone doesn't confirm the numerical factor 2π is correct — dimensional analysis can never verify dimensionless constants).

Solution Impulse, momentum, and force×time all share dimension [MLT⁻¹]. Force alone has dimension [MLT⁻²] — the odd one out.

Solution For a quantity of this power-law form, percentage errors combine as: %error in P = 2(%a) + 3(%b) + 1(%c) + ½(%d).

= 2(1) + 3(2) + 3 + ½(4) = 2 + 6 + 3 + 2 = 13%.