Count significant figures in given data:
Mass m = 5·580 kg → 4 significant figures (trailing zero after decimal is significant)
Side a = 9·0 cm = 9·0 × 10⁻² m → 2 significant figures
Calculate density:
V = a³ = (9·0 × 10⁻²)³ = 729 × 10⁻⁶ m³
ρ = m/V = 5·580 / (729 × 10⁻⁶) = 7654·3... kg/m³ = 7·6543 × 10³ kg/m³
Apply significant figures rule for division:
In multiplication/division: result has same sig figs as the least precise measurement.
Least sig figs = 2 (from side = 9·0 cm)
Round 7·6543 to 2 significant figures → 7·7
∴ ρ = 7·7 × 10³ kg/m³ → X = 7·7
ρ = m/a³ = 5.580/(729×10⁻⁶) = 7654 kg/m³
Rounded to 2 sig figs (from a = 9.0 cm) = 7.7 × 10³ kg/m³
📌 All non-zero digits are significant: 1234 → 4 sig figs
📌 Zeros between non-zero digits are significant: 1002 → 4 sig figs
📌 Leading zeros (before first non-zero digit) are NOT significant: 0·0012 → 2 sig figs
📌 Trailing zeros AFTER decimal point ARE significant: 9·0 → 2 sig figs; 5·580 → 4 sig figs
📌 Trailing zeros WITHOUT decimal point are ambiguous: 1200 could be 2, 3, or 4 sig figs
📌 Use scientific notation to be unambiguous: 1·2 × 10³ (2 sig figs); 1·200 × 10³ (4 sig figs)
The key principle: the result cannot be more precise than the least precise measurement used. For addition and subtraction: retain as many decimal places as the measurement with the fewest decimal places. Example: 3·125 + 1·7 = 4·825 → round to 4·8 (1 decimal place, same as 1·7). For multiplication and division: retain as many significant figures as the measurement with the fewest significant figures. Example: 2·5 × 3·14 = 7·85 → round to 7·9 (2 sig figs, same as 2·5).
Writing 9·0 (with decimal point and trailing zero) explicitly indicates the measurement was made to the nearest 0·1 cm — the zero is significant. If we wrote just "9", it would mean we only know the measurement to the nearest 1 cm (1 significant figure). By writing 9·0, the experimenter signals: "I measured this to be 9·0 cm, not 8·9 or 9·1." This distinction is crucial in scientific reporting. Similarly, 5·580 (not 5·58) tells us the fourth digit is 0, not unknown.
📌 Systematic error: Same direction every time — instrument calibration error, zero error, personal bias. Can be corrected if identified.
📌 Random error: Unpredictable fluctuations — reduced by taking multiple measurements and averaging.
📌 Gross error: Blunders — reading instrument wrongly, noting wrong value. Eliminated by careful observation.
📌 Least count error: Due to finite precision of instrument. Cannot be smaller than the least count of the measuring instrument.
Absolute error: Δa = |a_measured − a_true|. Mean absolute error: Δa_mean = (Δa₁ + Δa₂ + ... + Δaₙ)/n. Relative error: Δa/a (dimensionless). Percentage error: (Δa/a) × 100%. For derived quantities using multiplication/division: the relative errors add. Example: ρ = m/V = m/a³. Then Δρ/ρ = Δm/m + 3(Δa/a). The factor 3 comes from the power 3 on 'a' — errors in a get multiplied 3 times. This shows why accurately measuring linear dimensions is critical for volume calculations.
📌 Addition/Subtraction: Z = A ± B → ΔZ = ΔA + ΔB (absolute errors add)
📌 Multiplication/Division: Z = A×B or A/B → ΔZ/Z = ΔA/A + ΔB/B (relative errors add)
📌 Power: Z = Aⁿ → ΔZ/Z = n(ΔA/A) (relative error multiplied by power)
📌 Example: T = 2π√(L/g) → ΔT/T = ½(ΔL/L) + ½(Δg/g)
📌 Note: errors always ADD (never cancel) — worst-case analysis
Every physical quantity has dimensions expressed in terms of fundamental quantities: M (mass), L (length), T (time), A (current), K (temperature), mol (amount), cd (luminous intensity). Uses of dimensional analysis: (1) Check correctness of equations — dimensions must match on both sides. (2) Derive formulae — if you know which quantities are involved and how they relate dimensionally. (3) Convert units between systems. Limitations: dimensionless constants cannot be determined; cannot distinguish between scalar and vector quantities; doesn't work for trigonometric or logarithmic functions.
Accuracy refers to how close a measurement is to the true value — it relates to systematic error. Precision refers to how reproducible measurements are — it relates to random error. A measurement can be precise but inaccurate (e.g., consistently reading 2 cm too high due to a calibration error). A measurement can be accurate on average but imprecise (large scatter around the true value). Good experimental design aims for both: well-calibrated instruments (accuracy) AND multiple measurements with averaging (precision).