A-Level Physics: Quantities & Measurement Guide

TL;DR
Mastering units, errors and vectors is not "intro fluff" - it is the quality-control layer that guards every mark in kinematics, fields and practical Paper 4. This guide integrates IP (Sec 3/4) foundations with H2 exam technique: SI units → measurement → uncertainty → graphing → vectors.

Learning objectives (IP → H2 bridge)

By the end, you can:

1. Explain that every physical quantity = magnitude + unit.
2. Recall the 7 SI base quantities and use them to form derived units.
3. Use SI prefixes correctly (including the 2022 additions) and convert units swiftly.
4. Convert using standard form (scientific notation) and consistent significant figures.
5. Measure length (ruler, Vernier, micrometer) and time (stopwatch, pendulum) with zero-error checks.
6. Distinguish precision vs accuracy, and random vs systematic errors; quote uncertainties properly.
7. Record raw data, process with the right s.f./d.p. rules, and propagate uncertainties.
8. Present and interpret graphs (best-fit, gradient, intercept, error bars).
9. State precautions (safety + accuracy) and propose realistic improvements.
10. (H2 extension) Use dimensional homogeneity and vector decomposition confidently.

1 Physical quantities and SI units

A physical quantity is any measurable property (e.g., mass of a printer, area of a pool, speed of a bicycle). A valid measurement states both a number and a unit: example, mass = 12 kg (12 is the magnitude; kg is the unit). Missing units costs marks and blocks unit-checking.

1.1 SI base quantities

Mini-drill: Write the unit of gravitational field strength \( \mathrm{m\space s^{-2}} \).

1.2 Derived quantities and dimensional homogeneity

Build derived units from base units:

Dimensional check example
Verify \( s = ut + \tfrac{1}{2} a t^2 \).

• \([s] = \mathrm{m}\)
• \([ut] = \mathrm{m\space s^{-1}}\ \text{with}\ \mathrm{s} \rightarrow \mathrm{m}\)
• \([a t^2] = \mathrm{m\space s^{-2}}\ \text{with}\ \mathrm{s^2} \rightarrow \mathrm{m}\)

All terms reduce to metres.

2 SI prefixes (and how not to lose marks)

Why: Compact notation and fast conversions.
Exam cue: Do not mix decimal prefixes with powers of 2 (e.g., 1 kbit = 1000 bit, not 1024).

3 Precision, accuracy and error types

Precision = reproducibility (readings cluster tightly).
Accuracy = closeness to true value (cluster at the bullseye).

• Random errors (hurt precision): fluctuate in sign and magnitude (e.g., parallax, reaction time, background vibration). Reduce by repetitions and averaging.
• Systematic errors (hurt accuracy): constant bias each time (e.g., zero error, miscalibration). Detect via standards and correct arithmetically.

Language to use in answers
"Random error increases the scatter of repeated readings (poor precision) and is reduced by taking many readings and averaging."
"Systematic error shifts all readings in one direction (poor accuracy) and is corrected by subtracting or adding the zero error or recalibrating."

3.1 Significant figures vs decimal places

• Significant figures (s.f.): digits that convey precision. Rules: all non-zeros; zeros between non-zeros; trailing zeros only if a decimal point is present.
• Decimal places (d.p.): digits to the right of the decimal point, regardless of significance.

Recording rule of thumb

• Analogue instrument: uncertainty = ± 1/2 of the smallest division.
• Digital instrument: uncertainty = ± 1 least significant digit.

4 Scientific notation (standard form)

Express numbers as \( N \times 10^n \) with \( 1 \le N < 10 \) and integer \( n \).

• Move decimal left → \( n>0 \); move right → \( n<0 \).
• Significant figures live in \( N \).

Examples
5,600 = 5.6 x 10^3
9,020,000 = 9.02 x 10^6
0.0045 = 4.5 x 10^-3

5 Converting units (factor-label method)

Steps

1. Separate number and unit.
2. Choose conversion factor(s) so units cancel.
3. Multiply through.
4. Round to sensible s.f. (match the given data).

Speed

Volume

Density

Square/cube trap
If 1 m = 100 cm, then 1 m^2 = (100)^2 cm^2 = 10^4 cm^2 and 1 m^3 = (100)^3 cm^3 = 10^6 cm^3.

6 Measuring length (least counts, zero errors, readings)

6.1 Ruler (scale)

• Least count: 1 mm (0.1 cm).
• Method: align with zero; avoid parallax; estimate to half-division if needed.
• Quote: 12.35 cm ± 0.05 cm (± 1/2 least division).

6.2 Vernier calipers

• Typical least count: 0.10 mm = 0.01 cm.
• Reading:
  \( \text{Measurement} = \text{main scale} + (\text{Vernier coincidence}) \times \text{least count} \).
• Zero error: close jaws; if Vernier zero is right of main zero ⇒ +ve zero error; left ⇒ -ve.
  \( \text{True} = \text{Measured} - \text{Zero error} \).

Worked
Main = 2.30 cm; Vernier coincidence = 4; least count = 0.01 cm
Measured = 2.30 + 4(0.01) = 2.34 cm.
If zero error = +0.02 cm → True = 2.34 - 0.02 = 2.32 cm.

6.3 Micrometer screw gauge

• Least count: 0.01 mm (0.001 cm).
• Use: place between anvil and spindle; tighten with ratchet to avoid over-torque (backlash).
• Reading:
  \( \text{Measurement} = \text{Sleeve reading} + (\text{Thimble reading}) \times \text{least count} \).
• Zero error and correction as above.

Worked
Sleeve = 2.50 mm; Thimble = 13; least count = 0.01 mm
Measured = 2.50 + 13(0.01) = 2.63 mm.

7 Measuring time - the simple pendulum

Period \( T \): time for one full oscillation. For small angles (less than about 10 degrees),

\[ T = 2\pi \sqrt{\frac{L}{g}} \]

which implies a straight-line test:

\[ T^2 = \frac{4\pi^2}{g} L \]

Equipment: string, dense bob, rigid support, ruler/tape (for \( L \)), stopwatch, small-angle release.

Method (marks-friendly)

1. Measure \( L \) from pivot to the centre of mass of the bob.
2. Displace by a small angle; release without push; swing in one plane.
3. Time \( N = 20\text{ to }30 \) oscillations through the lowest point; repeat 3 to 5 trials.
4. Compute \( T = \frac{\text{total time}}{N} \); average across trials.
5. Plot \( T^2 \) (y) against \( L \) (x); gradient \( m = \frac{4\pi^2}{g} \) which gives \( g = \frac{4\pi^2}{m} \).

Common errors and mitigation

• Reaction time → time many oscillations; start/stop at mid-point.
• Large angle → increases period; keep less than about 10 degrees.
• Pivot friction or air drag → smooth pivot; dense small bob.
• Length mis-measured → measure to bob centre, not edge.
• Elliptical path → steady release in a single plane.

8 Recording and processing raw data

8.1 Tabulation rules

• First column = independent variable (increasing order, regular intervals).
• At least 5 sets for linear, 7 for curved relationships.
• Repeat dependent readings and average.
• Include units in headers (e.g., Force / N, Time / s).

8.2 Quoting uncertainties

• Analogue: ± 1/2 smallest division.
• Digital: ± 1 in last displayed digit.
• For a mean of repeats, quote range/2 or the instrument-based uncertainty, whichever is larger (school-lab convention).

8.3 s.f. rules for calculations

• Add/Subtract → answer follows the least d.p. among inputs.
• Multiply/Divide → answer follows the least s.f. among inputs.
• Functions (e.g., constants like \( g = 9.81\ \mathrm{m\space s^{-2}} \)) → treat constants as exact unless stated; round by the rule of the measured inputs.

8.4 Propagating uncertainties (H2)

For a general product or quotient with powers, add percentage uncertainties:

\[ \frac{\Delta Q}{Q} = \frac{\Delta A}{A} + 2\frac{\Delta B}{B} + \frac{\Delta C}{C} \quad \text{for} \quad Q=\frac{A B^2}{C} \]

For sums and differences, add absolute uncertainties.

WA hack: In Paper 4, a quick percentage-uncertainty estimate often suffices to justify the dominant source of error.

9 Graphing: presentation and interpretation

9.1 Pre-sketching

• Choose axes: x = independent, y = dependent.
• Use scales that fill at least half a page, based on 1-2-5-10 steps.
• Label with quantity (unit), e.g., Force (N), Time (s).

9.2 Plotting

• Plot neat crosses or circled dots; include error bars if uncertainties are significant.
• Draw a best-fit straight line or smooth curve (not dot-to-dot). Aim for symmetric scatter around the line.

9.3 Post-sketch interpretation

• Gradient with units (e.g., \(\mathrm{N\space m^{-1}}\) for a force-extension graph).
• Intercept with physical meaning if applicable.
• State result with uncertainty and appropriate s.f.

Spreadsheet tip (H2 → Paper 4)
Import data → XY scatter → trendline. Use =LINEST(Y, X, TRUE, TRUE) to obtain gradient ± SE. Round your final quoted value to match the least precise raw input.

10 Worked micro-example (Pendulum data → g)

Given (sample):

• \( L = 0.800 \pm 0.001\ \mathrm{m} \)
• Time for \( 20 \) swings: \( T_{20} = 35.80, 35.72, 35.90\ \mathrm{s} \)
• Mean \( T_{20} = 35.81\ \mathrm{s} \Rightarrow T = 35.81/20 = 1.7905\ \mathrm{s} \Rightarrow T^2 = 3.207\ \mathrm{s^2} \)
  Suppose best-fit gradient \( m = 3.98\ \mathrm{s^2\space m^{-1}} \).
  Then \( g = \frac{4\pi^2}{m} = \frac{39.478}{3.98} = 9.92\ \mathrm{m\space s^{-2}} \).
  Quote to sensible s.f. with uncertainty from scatter or LINEST.

11 Scalars and vectors (H2 extension)

• Vector = magnitude + direction.
• Tip-to-tail for geometric addition; components for algebra: \( F_x = F \cos \theta, \ F_y = F \sin \theta \).

12 Order-of-magnitude estimates (exam-savvy)

Use one-sig-fig anchors: classroom length ~ 8 m; human reaction time ~ 0.2 s. This prevents paralysis on open-ended items and keeps unit sense sharp.

13 Precautions (safety and accuracy)

13.1 Personal safety

• PPE: goggles for splashes or shards, gloves for hot or corrosive materials, lab coat.
• Handle glassware and electrics carefully; power off before circuit changes.
• Keep benches tidy; avoid trailing cables.

13.2 Experimental accuracy

• Calibrate or zero instruments (balances, meters, calipers).
• Parallax: eye level with scale or meniscus.
• Control variables: e.g., constant temperature for resistance experiments.
• Repeat and average to reduce random error.
• Environment: shield from drafts or vibration; allow equipment to settle.

14 Three WA timing rules (H2 practical mindset)

1. 1 mark ≈ 1.5 min (SEAB design).
2. Start data questions by writing units before numbers.
3. Show working to bank method marks even if arithmetic slips.

15 References

• SEAB H2 Physics 9478 syllabus (2026)
• BIPM - SI Brochure, 9th Ed.
• NIST - SI Prefixes (2022 update)
• NIST Technical Note 1297 — Guidelines for Evaluating and Expressing Uncertainty
• Physics Classroom - Vectors and Motion

16 Call-to-action

Last updated 14 Jul 2025. Next review when SEAB issues the 2027 draft syllabus.