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All revision notes Revision notes · Measurements & UnitsMeasurements & Units revision notes
SI units, prefixes, standard form, significant figures, errors and uncertainty, fair tests, tables and graphs: the measuring skills every physics paper marks. Each section follows one lesson: what you need to know, key terms, the equations with units, a worked example, common mistakes and an exam tip.
1 Foundations
What you need to know
SI Units
- A measurement is a number times a unit. Change the unit and the number changes with it; the quantity stays the same. A smaller unit gives a bigger number.
- The SI base units include the metre (m, length), kilogram (kg, mass), second (s, time), ampere (A, current) and kelvin (K, temperature). The kilogram is the only base unit with a prefix in its name.
- Every other unit is a derived unit, built from base units: the newton is 1 kg m/s², the joule is 1 N m.
Prefixes
- A prefix multiplies a unit by a power of ten: G = 109, M = 106, k = 103, c = 10−2, m = 10−3, μ = 10−6, n = 10−9.
- Changing the prefix moves the decimal point; the amount stays the same. 2.0 g = 0.0020 kg; 4.3 ms = 4300 μs.
- At IGCSE, k, c and m are needed by every candidate; G, μ and n are Supplement.
Reading a Ruler
- A length is the reading at one end minus the reading at the other. A ruler need not start at zero.
- The smallest mark sets the resolution: 1 mm on a school ruler. Record a reading to that resolution, for example 5.8 cm, not 5.8000 cm.
- Look straight at each mark, at eye level, to avoid parallax error.
Scalars & Vectors
- A scalar has size only (distance, speed, time, mass, energy, temperature). A vector has size and direction (displacement, velocity, acceleration, force, weight, momentum, field strength).
- Vectors in the same line add or subtract: a plane at 60 m/s with a 25 m/s tailwind moves at 85 m/s, into a 25 m/s headwind at 35 m/s.
- Two vectors at right angles combine by Pythagoras for the size and tan θ for the direction, or by a scale drawing.
Key terms
- Base unit
- one of the SI units every other unit is built from, such as the metre or the second
- Derived unit
- a unit made from base units, such as the newton (kg m/s²)
- Resolution
- the smallest change an instrument can show, such as 1 mm on a ruler
- Scalar
- a quantity with size only
- Vector
- a quantity with size and direction
Equations
| Equation | What it says, symbols and units |
|---|---|
G = 109 M = 106 k = 103 c = 10−2 m = 10−3 μ = 10−6 n = 10−9 | Each prefix is a power of ten; neighbours from n to G are 1000 apart, so the decimal point moves three places. |
L = x2 − x1 | A length is the reading at the far end minus the reading at the near end.
|
R = √(F12 + F22) | The resultant of two forces (or velocities) at right angles, by Pythagoras; or find it with a scale drawing.
|
Diagram to know
A vector triangle: a plane's velocity through the air drawn north, the wind drawn east from its tip, and the resultant velocity over the ground from the start of the first arrow to the end of the second, with the angle θ east of north marked.
Worked example
A bird flies due north at 54 km/h through the air. A wind of 5.0 m/s blows from the west. Find the bird's speed over the ground in m/s, and its direction.
- Same units first: 54 km/h ÷ 3.6 = 15 m/s north. The wind blows towards the east, at right angles to it.
- v = √(15² + 5.0²) = √250 = 15.8 m/s.
- tan θ = 5.0 ÷ 15, so θ = 18° east of north.
Answer: 15.8 m/s (16 m/s to 2 s.f.) at 18° east of north
Common mistakes
- Writing a number with no unit. Without a unit a measurement means nothing.
- Moving the decimal point the wrong way. A bigger unit needs a smaller number: 2.0 g = 0.0020 kg, not 2000 kg.
- Reading only the right-hand end of a ruler that does not start at zero.
- Saying "wind from the west" blows west. It blows from the west towards the east.
Exam tip
Before you put any number into an equation, convert it to the base SI unit (km to m, g to kg, ms to s). Then the answer comes out in the SI unit too, and you cannot lose the unit mark.
AP Physics and NGSS: AP Physics 1 does not examine prefixes or base units as content, but every answer must carry correct SI units, and vectors are split into x and y components from the start (Unit 1).
2 Numbers in Physics
What you need to know
Standard Form & Orders of Magnitude
- Standard form writes a number as A × 10n, with A from 1 up to just under 10. The power n counts how many places the decimal point moved: negative for numbers below 1.
- 0.000 08 m = 8 × 10−5 m; 12 700 000 m = 1.27 × 107 m. A calculator shows 3E8 for 3 × 108.
- To compare two numbers in standard form, compare the powers first: 2 × 10−3 is bigger than 3 × 10−4.
- An order of magnitude is the nearest power of ten. The Earth, 1.27 × 107 m across, is of order 107 m.
Significant Figures
- An answer keeps as many significant figures as the least precise reading used: 8.6 cm × 5.4 cm = 46 cm², not 46.44 cm².
- Zeros in front only place the point and never count (0.0052 has two significant figures). A zero at the end after the decimal point was measured and counts (7.00 has three).
- The instrument sets the digits; the calculator just prints all it has. Round once, at the end.
Converting Units
- Convert units by multiplying by a fraction equal to one, such as 1000 m ÷ 1 km or 1 h ÷ 3600 s. So 72 km/h × 1000 ÷ 3600 = 20 m/s.
- A squared unit squares the factor and a cubed unit cubes it: 1 m² = 10 000 cm², 1 m³ = 1 000 000 cm³.
- 1 g/cm³ = 1000 kg/m³. Check the size after converting: a bigger unit needs a smaller number.
Key terms
- Standard form
- a number written as A × 10ⁿ, with A at least 1 and less than 10
- Order of magnitude
- the power of ten nearest to a quantity
- Significant figures
- the digits of a number that carry meaning about its precision
- Conversion factor
- a fraction equal to one, used to change a unit without changing the quantity
Equations
| Equation | What it says, symbols and units |
|---|---|
x = A × 10n, 1 ≤ A < 10 | Standard form: a front number from 1 to just under 10, times a power of ten.
|
1 m/s = 3.6 km/h | Divide km/h by 3.6 to get m/s; multiply m/s by 3.6 to get km/h. |
1 m2 = 104 cm2, 1 m3 = 106 cm3 | Square or cube the length factor (100) for areas and volumes. |
Worked example
A red blood cell is 7.5 × 10−6 m across. How many cells fit side by side in a row 3.0 mm long? Give the answer in standard form to two significant figures.
- Same unit first: 3.0 mm = 3.0 × 10−3 m.
- Number of cells = 3.0 × 10−3 ÷ 7.5 × 10−6 = 400.
- Both readings have two significant figures, so the answer is 4.0 × 102 cells.
- Check the size: 10−3 ÷ 10−6 = 103, and 3.0 ÷ 7.5 = 0.40, so 0.40 × 103 = 400 is the right size.
Answer: 4.0 × 102 cells
Common mistakes
- Counting the zeros in front as significant: 0.0450 has three significant figures, not five.
- Copying every calculator digit into the answer, or rounding in the middle of a calculation.
- Using the length factor for an area or a volume: 1 m³ is 106 cm³, not 100 cm³.
- Writing the front number outside 1 to 10, such as 45 × 103, and calling it standard form.
Exam tip
When a question asks for an answer "to an appropriate number of significant figures", look at the data: give the same number as the least precise value in the question, usually two or three. Writing ten digits loses the mark even if the physics is right.
AP Physics and NGSS: AP Physics 1 expects order-of-magnitude estimates and sensible significant figures in every free-response answer but does not test them on their own; NGSS treats scale and quantity as a crosscutting concept rather than a standard.
3 Measurement & Error
What you need to know
Measuring Cylinders
- To read a scale, first find what one division is worth: (difference between two labels) ÷ (number of gaps between them). Then count divisions up from the label below.
- Read a liquid level at the bottom of the meniscus, with your eye level with the surface.
- Use the smallest measuring cylinder that holds the liquid: its marks are finest. A beaker's marks are only a rough guide. 1 cm³ = 1 ml and 1000 cm³ = 1 litre.
Measuring Multiples
- To time something too quick to time once, time many and divide: period T = total time ÷ number of oscillations. One oscillation is there and back.
- Reaction time, about 0.2 s, rides on every start and stop. Timing 10 oscillations shares that error ten ways.
- Start and stop timing as the pendulum passes the centre, where it moves fastest. The same trick works for thin things: measure a stack of sheets or a coil of wire, then divide.
Random & Systematic Error
- Random errors scatter readings both ways. Taking the mean of repeats reduces their effect.
- Systematic errors shift every reading the same way, such as a zero error. Repeating never removes them; only correcting the instrument or the method does. True value = reading − zero error.
- Precise readings are close together; accurate readings are close to the true value. Readings can be precise but not accurate.
Percentage Error
- Percentage error = |measured − accepted| ÷ accepted × 100%. Divide by the accepted value, not by your own answer.
- If the accepted value is zero a percentage error means nothing; quote the difference instead.
Key terms
- Meniscus
- the curved surface of a liquid in a tube; read the bottom of it
- Random error
- an error that scatters readings unpredictably either side of the true value
- Systematic error
- an error that shifts every reading by the same amount or in the same direction
- Zero error
- a non-zero reading when the true value is zero
- Accurate
- close to the true value
- Precise
- repeated readings close to each other
Equations
| Equation | What it says, symbols and units |
|---|---|
T = tn | Time many oscillations and divide to find one; the timing error is shared out too.
|
true value = reading − zero error | Correct a systematic zero error by subtracting it from every reading. |
% error = |measured − accepted|accepted × 100% | How far a result is from the accepted value, as a percentage of that value. |
Worked example
(a) A student times 20 swings of a pendulum as 32.8 s. The accepted period is 1.60 s. Find the measured period and the percentage error. (b) A micrometer reads 0.02 mm with its jaws closed. It then reads 1.27 mm on a wire. What is the wire's true diameter?
- (a) T = 32.8 ÷ 20 = 1.64 s.
- Gap = |1.64 − 1.60| = 0.04 s, so % error = 0.04 ÷ 1.60 × 100 = 2.5 %.
- (b) The zero error is +0.02 mm, a systematic error on every reading: true diameter = 1.27 − 0.02 = 1.25 mm.
Answer: (a) 1.64 s, 2.5 % (b) 1.25 mm
Common mistakes
- Reading the top of the meniscus, or reading it from above or below instead of at eye level.
- Timing one swing with a stopwatch and calling it accurate. Reaction time is a large part of a short event.
- Saying repeats remove a systematic error. They only reduce random error.
- Mixing up precise and accurate.
Exam tip
When asked how to improve a timing, give the method and the reason: "time 20 oscillations and divide by 20, so the reaction-time error is shared over 20 swings." A method with no reason usually scores only half.
AP Physics and NGSS: AP Physics 1 assesses error analysis as a science practice in experimental-design questions: you are expected to name a source of systematic error and say which way it shifts the result.
4 Planning an Investigation
What you need to know
Variables & a Fair Test
- The independent variable is the one you change on purpose; the dependent variable is the one you measure; control variables are kept the same.
- A fair test changes one variable at a time, so the result is valid: it tests what it says it tests.
- A hypothesis is an idea you can test; a prediction says what you expect to measure if it is right.
Instruments, Range & Safety
- Choose an instrument whose resolution is small compared with the value and whose range covers it. A micrometer (0.01 mm) for a wire; a tape measure for a corridor; a light gate, not a stopwatch, for a 0.30 s event.
- Plan at least five values of the independent variable, evenly spaced, over as wide a range as is safe.
- Name each hazard with its precaution: a stop block for a runaway trolley, a sand tray under falling masses.
Uncertainty from Repeats
- Repeat each reading and take the mean. The uncertainty of the mean is about half the range of the repeats.
- Write the result as (mean ± uncertainty) with the unit after the bracket, both numbers to the same decimal places: (2.44 ± 0.06) s.
- Percentage uncertainty = uncertainty ÷ value × 100%. A reading from a scale is uncertain by at least half its resolution.
Anomalies & Agreement
- An anomaly is a result far from the others. Check it and repeat it; do not average it in.
- Results are repeatable if the same person and method give close values, and reproducible if another person or method does too.
- Two results agree if their ranges (value − uncertainty to value + uncertainty) overlap. With no uncertainty given, IGCSE treats results within 10% of each other as the same.
Key terms
- Independent variable
- the variable you change
- Dependent variable
- the variable you measure
- Control variable
- a variable kept the same so the test is fair
- Uncertainty
- the range within which the true value is expected to lie
- Anomaly
- a result that does not fit the pattern of the others
- Reproducible
- giving similar results when repeated by another person or with another method
Equations
| Equation | What it says, symbols and units |
|---|---|
mean = sum of the readingsnumber of readings | Average the repeats, leaving out any anomaly. |
uncertainty ≈ largest − smallest2 | Half the range of the repeated readings. |
% uncertainty = uncertaintyvalue × 100% | The uncertainty as a share of the value; smaller is better. |
Worked example
Four timings of a ball rolling down a ramp are 2.31 s, 2.37 s, 2.35 s and 2.33 s. Write the result with its uncertainty, find the percentage uncertainty, and decide whether it agrees with another group's (2.30 ± 0.02) s.
- Mean = (2.31 + 2.37 + 2.35 + 2.33) ÷ 4 = 9.36 ÷ 4 = 2.34 s.
- Uncertainty = (2.37 − 2.31) ÷ 2 = 0.03 s, so the result is (2.34 ± 0.03) s.
- % uncertainty = 0.03 ÷ 2.34 × 100 = 1.3 %.
- Ranges: 2.31 s to 2.37 s, and 2.28 s to 2.32 s. They overlap, so the results agree.
Answer: (2.34 ± 0.03) s; 1.3 %; the two results agree
Common mistakes
- Changing two variables at once, so the cause of any change cannot be known.
- Calling the variable you measure the independent variable.
- Writing the mean and the uncertainty to different decimal places, such as (2.436 ± 0.06) s.
- Averaging an anomaly into the mean instead of checking and repeating it.
Exam tip
In a planning question, use the words independent, dependent and control and name each one for that experiment, with its unit. Then say how many values, over what range, how many repeats, and one hazard with its precaution. Each of these is usually a separate mark.
AP Physics and NGSS: AP Physics 1 free-response questions often ask you to design an experiment: name the measured quantities, the equipment, and which graph you would plot to get a straight line. NGSS covers the same skills as the practice "planning and carrying out investigations".
5 Tables & Graphs
What you need to know
Tables, Axes & Plotting
- In a table, give each column heading the quantity and its unit once (load / N); the cells hold plain numbers, all to the same number of decimal places.
- Put the independent variable along the bottom (x-axis) and the dependent variable up the side (y-axis), with labels and units.
- Choose a scale that goes up in 1, 2 or 5 (or 10, 20, 50) per square and uses more than half the grid both ways. Plot each point as a neat cross to within half a small square.
Lines of Best Fit & Proportion
- A line of best fit is one thin straight line or smooth curve with the points spread evenly either side. It need not pass through any point; leave anomalies out.
- Two quantities are directly proportional when their graph is a straight line through the origin. A straight line that misses the origin is linear but not proportional.
- A straight line has the equation y = mx + c, where m is the gradient and c the y-intercept.
Gradient, Intercept & Tangents
- The gradient = rise ÷ run = Δy ÷ Δx, and it has a unit (unit of y ÷ unit of x). Use a large triangle, over at least half the line.
- On a curve, the gradient at a point is the gradient of the tangent drawn there: the rate of change at that moment.
- Reading between plotted points is interpolation; extending the line beyond them, for example back to an axis, is extrapolation. The intercept is where the line meets the y-axis.
Key terms
- Line of best fit
- a line drawn so the points are evenly spread either side of it
- Directly proportional
- a straight-line graph through the origin: double one, double the other
- Gradient
- the change in y divided by the change in x, with its unit
- Intercept
- the value where a line crosses the y-axis
- Tangent
- a straight line that touches a curve at one point and has the curve's slope there
Equations
| Equation | What it says, symbols and units |
|---|---|
m = ΔyΔx | Gradient of a straight line or of a tangent: the rise divided by the run.
|
y = mx + c | A straight line; c = 0 means y is directly proportional to x.
|
Diagram to know
A straight-line graph with a large gradient triangle drawn on it, covering more than half the line, with Δy and Δx labelled and read from the line, not from data points. Beside it, a curve with a tangent touching it at one point.
Worked example
A burning candle's length–time graph is a straight line through (5.0 min, 13.6 cm) and (45.0 min, 10.4 cm). Find the gradient, the length at t = 0, and how long the candle takes to burn away completely.
- Δy = 10.4 − 13.6 = −3.2 cm; Δx = 45.0 − 5.0 = 40 min.
- Gradient = −3.2 ÷ 40 = −0.080 cm/min: the candle loses 0.080 cm each minute.
- Extrapolate back 5.0 min from the first point: length at t = 0 = 13.6 + 0.080 × 5.0 = 14.0 cm.
- Time to burn away = 14.0 ÷ 0.080 = 175 min (an extrapolation, so it assumes the rate stays the same).
Answer: −0.080 cm/min; 14.0 cm; 175 min
Common mistakes
- Writing units in every cell of a table instead of once in the heading.
- Choosing an awkward scale (such as 3 or 1.75 per square) or one that uses less than half the grid.
- Joining the points dot to dot instead of drawing one line of best fit.
- Calling any straight line proportional. It must also pass through the origin.
- Using a small gradient triangle, or reading it from data points that are off the line.
Exam tip
When you find a gradient, draw the triangle on the graph and write the two readings you used. Examiners give marks for a large triangle and for the unit of the gradient, so a correct number alone can lose two marks.
AP Physics and NGSS: AP Physics 1 goes further with linearisation: choosing what to plot (for example T² against L for a pendulum) so the graph is a straight line whose gradient gives a constant. NGSS treats the same skills as "analysing and interpreting data".
Written in Physiworld's own words. The course tags come from the same mapping as the syllabus maps; equations are on the IGCSE, GCSE, AP Physics 1 and NGSS formula sheets. Values use g = 9.8 N/kg.
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