1.2 Measurement — American System Physics Revision Notes & Topical Questions

Part 1 of 7Key ideas

1

Mean = sum of readings ÷ number of readings.

2

Percentage error = |measured value − accepted value| ÷ accepted value × 100%.

3

Uncertainty of a single reading ≈ ± half the smallest scale division (e.g. ±0.5 mm on a mm rule).

4

Uncertainty from repeats ≈ ± half the range = ±(largest − smallest) ÷ 2.

5

Measuring many identical items together reduces uncertainty: thickness of one sheet = thickness of a stack ÷ number of sheets.

6Timing many oscillations

Timing many oscillations period T = total time ÷ number of oscillations, reducing the effect of reaction time (~0.2 s).

7

Corrected reading = observed reading − zero error (a negative zero error is therefore added).

8

Volume of irregular solid = final water level − initial water level (displacement), in cm³ or m³.

Part 2 of 7Learning objectives

1

Distinguish between precision and accuracy.

2

Identify random and systematic errors and suggest ways to reduce them.

3

Record measurements with appropriate units, resolution and uncertainty.

4

Use repeat readings, multiple-item and multiple-oscillation methods to improve measurements.

Part 3 of 7Essential knowledge

1

1.2 Measurement concept map: Timing many oscillations, Parallax, Measurement, Precision, Accuracy — Visual guide to the key ideas explained in this lesson.

2

All measurements are comparisons with a standard; because instruments have limited resolution, no measurement is exact.

3

A metre rule has resolution 1 mm; a vernier caliper 0.1 mm; a micrometer screw gauge 0.01 mm; a digital stopwatch 0.01 s (but human reaction time of about 0.2–0.3 s dominates).

4

Precision depends on the instrument and the spread of repeated readings. Accuracy depends on how close the result is to the accepted value; a precise instrument that is badly calibrated can still be inaccurate.

5

Record a reading to the resolution of the instrument: on a mm-scale ruler write 12.0 cm, not 12 cm, to show the reading is precise to 0.1 cm.

6

Random errors come from unpredictable changes (air currents, reaction time, estimating between scale marks). Averaging many readings reduces their effect.

7

Systematic errors bias every reading in one direction (zero error, stretched tape, meter reading 0.2 A with no current). They are detected by calibration and corrected, not averaged away.

8Parallax

Parallax reading a scale or meniscus from above or below gives a false value. Read liquid volumes at the bottom of the meniscus, at eye level.

9

To measure a small quantity accurately, measure many together: e.g. the thickness of 100 sheets of paper, or the time for 20 swings of a pendulum, then divide.

10

Volume of a regular solid is found from measured dimensions (cuboid V = l × w × h); an irregular solid by displacement of water in a measuring cylinder.

11

Anomalous results are readings that do not fit the pattern; they should be checked, repeated and, if justified, excluded from the mean.

12

Percentage error compares the result to an accepted value; a small percentage error means an accurate result.

Part 4 of 7Process & sequence

1

Choose an instrument whose resolution suits the size of the quantity (e.g. vernier caliper for a 2 cm diameter, tape measure for a 10 m length).

2

Check for and record any zero error before starting.

3

Position your eye level with, and perpendicular to, the scale to avoid parallax error.

4

Take the reading to the resolution of the instrument and record it with its unit and uncertainty.

5

Repeat the measurement at least three times, discarding anomalous values.

6

Calculate the mean and correct it for any zero error.

7

Compare with the accepted value using percentage error to judge accuracy.

Improving a timing measurementwrite these steps in this order; each step is a marking point.

1

Start the stopwatch as the pendulum passes a fixed marker at the centre of the swing.

2

Count 20 complete oscillations (there and back).

3

Stop the watch and record the total time to 0.01 s.

4

Repeat three times and calculate the mean total time.

5

Divide the mean by 20 to obtain the period with a much smaller percentage uncertainty.

Part 5 of 7Worked examples

1.57 s

Five students time 20 swings of a pendulum: 31.2 s, 31.6 s, 31.4 s, 31.0 s, 31.3 s. Find the mean period and its uncertainty.

Mean time for 20 swings = (31.2 + 31.6 + 31.4 + 31.0 + 31.3)/5 = 156.5/5 = 31.3 s.
- Period T31.3/20 = 1.57 s.
- Uncertainty in totalhalf range = (31.6 − 31.0)/2 = ±0.3 s, so uncertainty in T = 0.3/20 = ±0.015 s (≈ ±0.02 s).

2.8%

A student measures g as 9.54 m/s². The accepted value is 9.81 m/s². Calculate the percentage error.

Percentage error = |9.54 − 9.81| / 9.81 × 100% = 0.27/9.81 × 100% = 2.8% (2 s.f.).

15.4 cm

A ruler has a zero error of +0.2 cm. It reads 15.6 cm for a pencil. What is the true length?

Corrected reading = observed − zero error = 15.6 − 0.2 = 15.4 cm.

Part 6 of 7Summary

1

Every measurement has an uncertainty; state it as value ± uncertainty.

2

Precision is how close repeated readings are; accuracy is how close to the true value.

3

Random errors are reduced by repeating and averaging; systematic errors are not.

4

Avoid parallax by reading scales at eye level, perpendicular to the mark.

Part 7 of 7Compare: Compare

Precision vs accuracy
Situation	Precise?	Accurate?
Readings close together and close to true value	Yes	Yes
Readings close together but all far from true value (e.g. zero error)	Yes	No
Readings widely spread but mean near true value	No	Yes (on average)
Readings widely spread and far from true value	No	No

Types of error
Error	Cause	Effect	How to reduce
Random	Reaction time, fluctuating conditions, estimating between marks	Scatter either side of true value	Repeat and average; use data loggers
Systematic	Zero error, badly calibrated meter, consistent parallax	All readings too high or too low	Calibrate, correct for zero error, improve method

Key terms

Command terms

Identify / Describe (FRQ)

A short, specific statement; describing requires relevant characteristics, not a single word.

Explain / Justify (FRQ)

Give the cause-and-effect reasoning or evidence; a restated fact does not score.

Calculate

Show setup, substitution and the answer with units.

Predict

State the expected outcome; add a reason only if asked to justify.

MCQ technique

Read every option; eliminate distractors that are true statements but do not answer the question; watch for NOT/EXCEPT.

Exam technique

Keywords the mark scheme looks for in this lesson

Measurement · Precision · Accuracy · Resolution · Uncertainty · Random error · Systematic error · Zero error · Parallax error · Calibration · Mean (average) · Percentage error · Reliable (repeatable) result

Learn each definition in the Key words tab exactly; a missing keyword in a definition usually loses the mark.

How to get full marks (examiner guidance)

1

Answer exactly what is asked; extra correct material does not earn credit, and contradictory statements can cancel credit (AP FRQ).

2

Use precise vocabulary, never “it/stuff/things”.

3

Show units and working in calculations.

4

In MCQs, predict the answer before reading the options, then eliminate.

Words to avoid

1

repeating removes errorsrepeating and averaging reduces the effect of random errors.

2

accurate means the readings are close togetherprecise means close together; accurate means close to the true value.

3

human errorthe specific error, e.g. reaction time when starting the stopwatch.

4

repeating the experiment fixes a zero errorsubtract the zero error from each reading.

5

read the scale carefullyread with the eye level with and perpendicular to the scale to avoid parallax.

Measurement — American System Physics Revision Notes & Topical Questions