Full chapter
Physical quantities and measurement
All 8 topics and the revision summary on one page.
01
Physical quantities and units
A physical quantity describes something measurable, such as a length, a time or a mass. Its unit tells us what the number means.
25 cm and 0.25 m describe the same length. Changing the unit changes the numerical value, but does not change the object. Writing just "25" would leave the length unclear: 25 millimetres, centimetres and metres are very different.
A physical quantity is usually expressed as a numerical value and a unit. Some quantities are dimensionless: for example, a ratio of two lengths has no unit because the length units cancel.
Base quantities and SI units
SI is the international system of units. Use these six base quantities and their SI units in this course. Other quantities can be formed by combining base quantities.
| Quantity | SI unit | Unit symbol |
|---|---|---|
| Length | metre | m |
| Mass | kilogram | kg |
| Time | second | s |
| Electric current | ampere | A |
| Thermodynamic temperature | kelvin | K |
| Amount of substance | mole | mol |
Write the kelvin as K, without a degree sign. The SI base unit of mass is the kilogram, kg, even though the gram is also a familiar mass unit. One kilogram is 1000 g.
A quantity symbol is different from a unit symbol
In m = 2.0 kg, the first m is a symbol for the quantity mass; kg is its unit. In l = 2.0 m, lowercase l represents length and m is the unit metre. Read the meaning given in the question instead of deciding what a letter means on its own.
Worked example
Getting a unit from a calculation
A trolley travels a total distance of 2.4 m in 0.80 s. Let d represent the total distance, t the elapsed time and v the average speed.
v = d / t = 2.4 m / 0.80 s = 3.0 m/s
Dividing distance by time gives metres per second. The result means that travelling steadily at 3.0 metres each second would cover the same distance in the same time. The unit follows from the quantities in the calculation.
Make a guess. You are not marked.
You want the period of a pendulum, the time for one complete swing, as accurately as possible with a stopwatch. Which is better: timing one swing, or timing twenty swings and dividing by twenty?
Show the answer
Time twenty swings and divide by twenty.
Your reaction time error is about the same either way, about 0.2 s. Spread over twenty swings it becomes 0.01 s per swing, instead of 0.2 s on a single swing.
For more journeys, see distance, speed and velocity. To change the units before calculating, use prefixes and physical scales.
02
Prefixes and physical scales
A prefix changes the size of a unit. Milli means one thousandth, so a millimetre is one thousandth of a metre.
The numerical value can change while the physical quantity stays the same. See quantities and units if that distinction is unfamiliar.
Powers of ten make very large and small numbers easier to write. 103 = 1000, while 10-3 = 1 / 1000 = 0.001. A negative exponent means a fraction here; it does not mean a negative length or current.
| Prefix | Symbol | Factor |
|---|---|---|
| nano | n | 10-9 |
| micro | µ | 10-6 |
| milli | m | 10-3 |
| centi | c | 10-2 |
| deci | d | 10-1 |
| kilo | k | 103 |
| mega | M | 106 |
| giga | G | 109 |
| tera | T | 1012 |
Capital letters matter: mA means milliamperes, but MA means megaamperes. The prefix kilo is lower-case k; the upper-case K is the unit kelvin.
Worked examples
Convert by replacing the prefixed unit
- Millimetres to metres: 1 mm = 0.001 m, so 3.2 mm = 3.2 x 0.001 m = 0.0032 m.
- Milliamperes to amperes: 1 mA = 0.001 A, so 250 mA = 250 x 0.001 A = 0.250 A.
- Metres to millimetres: 1 m = 1000 mm, so 0.0045 m = 0.0045 x 1000 mm = 4.5 mm.
Check the direction: expressing a length in a smaller unit needs a larger number of those units. There are 1000 millimetres in each metre.
Try it step by step
Work it out in steps: Length in m, then Width in m, then Area in m2.
Show the answer
Problem 1.
Length in m: Length = 25 x 0.01 = 0.25 m
Width in m: Width = 40 x 0.001 = 0.040 m
Area in m2: Area = 0.25 x 0.040 = 0.010 m2 (2 sf)
Problem 2.
Length in m: Length = 60 x 0.01 = 0.60 m
Width in m: Width = 25 x 0.001 = 0.025 m
Area in m2: Area = 0.60 x 0.025 = 0.015 m2 (2 sf)
Problem 3.
Length in m: Length = 4.0 x 0.01 = 0.040 m
Width in m: Width = 15 x 0.001 = 0.015 m
Area in m2: Area = 0.040 x 0.015 = 0.00060 m2 (2 sf)
Area and volume need more than one conversion
A square measuring 1 cm along each side has an area of 1 cm2. Each side is 0.01 m, so convert both dimensions when calculating its area.
The same reasoning applies to a cube: 1 cm3 = (0.01 m)3 = 0.000001 m3. Apply the length conversion to all three dimensions.
Does the size of the answer make sense?
An order of magnitude gives the approximate scale of a quantity as a power of ten. For example, 100 m is 1 m. The objects below vary in size; these values are useful reference scales, not exact measurements of every example.
| Object and dimension | Approximate scale |
|---|---|
| Diameter of an atom | 10-10 m |
| Width of a bacterium | 10-6 m |
| Width of a human hair | 10-4 m |
| Diameter of a coin | 10-2 m |
| Height of a person | 100 m |
| Diameter of Earth | 107 m |
If a calculation gives 0.02 m for the width of a human hair, pause: 0.02 m is 2 cm, about the width of a small coin. A hair is much thinner, with a width of roughly 10-4 m. Check the original reading and conversion before accepting the answer.
Check your understanding
A current is 250 mA. Which conversion to amperes is correct?
Show the answer
0.250 A
1 mA = 0.001 A, so 250 mA = 250 x 0.001 A = 0.250 A. The current has not changed; only its unit has.
03
Choosing measuring instruments
Make a guess. You are not marked.
You need the diameter of a thin wire, about 0.5 mm across. Which instrument gives the most precise reading?
Show the answer
A micrometer screw gauge, which reads to 0.01 mm.
A micrometer reads to 0.01 mm, so a 0.5 mm diameter is measured to two more figures than a ruler could give. Vernier calipers read only to 0.1 mm.
Choose an instrument that can cover the measurement and distinguish the detail you need.
A metre rule is useful for a book, but its divisions are too coarse to measure the thickness of one sheet of paper well. A micrometer can resolve a small thickness, but cannot directly span a room.
- Range
- The span of values the instrument can measure, such as 0 to 150 mm for a pair of calipers.
- Resolution
- The smallest change you can read from the scale or display, such as a 0.01 mm display step.
- Precision
- How closely repeated measurements agree. Readings with little spread are precise, but may all share the same error.
- Accuracy
- How close a measurement is to the true or accepted value. Checking the zero and using the instrument correctly help avoid a consistent offset.
A smaller display step does not guarantee an accurate measurement. A digital balance could show the same value repeatedly while its zero is wrong. Its readings would agree closely but remain displaced from the true value.
Match the instrument to the object
These are example ranges and resolutions. Check the actual instrument's markings and specification before choosing it.
Digital calipers may also have inside jaws for an internal diameter and a depth rod for a hole. Choose the correct contact surfaces. For a diameter, align the measurement across the widest part through the centre; tilting the instrument can give the wrong length.
Ask three questions before measuring
- Will it fit? Check the object's expected size against the instrument's range.
- Can I distinguish enough detail? Check the divisions or display step against the required measurement.
- Can I use it properly? Consider alignment, access, the object's shape and whether contact could deform it.
For a very thin sheet, another useful approach is to measure the thickness of a closely packed stack and divide by the number of sheets. This gives an average thickness; gaps or compressing the stack would affect the result.
Once an instrument is chosen, use sensible measurement methods to read it, check its zero and record the result.
Check your understanding
A rod is about 120 mm long. You need an instrument with a reading step of 0.1 mm or smaller. Which is suitable for measuring its length directly?
Show the answer
Digital calipers with a 0-150 mm range and 0.01 mm resolution.
120 mm is within the 150 mm range, and a 0.01 mm reading step is smaller than 0.1 mm. You must still check the zero and align the jaws correctly; the display step does not guarantee accuracy.
04
Making useful measurements
Make a guess. You are not marked.
You want the thickness of one sheet of paper using a ruler marked in millimetres. What is the best method?
Show the answer
Measure a stack of 100 sheets and divide the thickness by 100.
One sheet is far thinner than 1 mm, so a ruler cannot read it. A stack of 100 is several millimetres thick, and dividing by 100 gives one sheet.
A useful measurement depends on how you take the reading, not just which instrument you choose.
Choose a suitable range and resolution first. Then look for a specific source of error and a method that addresses it.
Read the difference between the ends
Place a rule parallel and close to the length being measured. If the object does not start at zero, read both ends and subtract. This is also useful when a rule's zero end is worn.
Look perpendicularly at the scale. If the object and markings are at slightly different heights, looking from one side makes them appear to line up differently. This is parallax error. Keep the object close to the scale as well as placing your eye correctly.
Check the zero before trusting the display
Clean and gently close the jaws of digital calipers, then check the reading. If they correctly close with no object between them but the display is not zero, there is a zero offset. Use the instrument's zero control when appropriate, or correct a stated stable offset.
Worked example
Correcting a positive zero error
Closed calipers read +0.02 mm. With an object between the jaws, they read 3.42 mm. Assume the zero offset stays the same.
Corrected reading = displayed reading - zero offset
= 3.42 - (+0.02) = 3.40 mm
The instrument adds 0.02 mm even when the true gap is zero, so subtract that extra amount. The same rule uses the sign of the offset: subtracting a negative zero offset would increase the reading.
Repeating a reading does not remove a stable zero error. Repeats help you judge and reduce random variation through averaging. A consistent offset needs a correction or a properly adjusted instrument.
Time several complete oscillations
The period of a pendulum is the time for one complete oscillation. The bob must return to the same reference point moving in the same direction. Going from one side to the other is only half an oscillation.
- Place a fixed reference mark beside the path, for example at the centre of the swing.
- Start timing as the bob passes the mark in a chosen direction. Count this as zero.
- Count one each time it next passes the mark in that same direction. Stop at 20.
- Repeat the timing under the same conditions, find the mean elapsed time, then divide by 20.
| Trial | Time for 20 oscillations / s |
|---|---|
| 1 | 40.6 |
| 2 | 40.2 |
| 3 | 40.4 |
The delay in reacting at the start and finish is a much smaller fraction of 40 s than of 2 s. Timing many oscillations reduces its fractional effect on the period. Repeating and averaging helps with variation in those delays, but cannot guarantee that all timing errors disappear.
The calculated 2.02 s does not mean a person directly timed one swing to the nearest 0.01 s. It comes from dividing a longer measured interval by an exact count of oscillations.
Use displaced water to measure an irregular solid
Choose a measuring cylinder large enough for the object and final water level, with divisions fine enough to distinguish the rise. Put it upright on a level surface. For water, read the bottom of the curved meniscus at eye level.
This works when the solid is fully submerged, does not dissolve or absorb water, and no water is lost. Remove trapped air bubbles: they displace extra water and would make the calculated solid volume too large. Avoid splashing when lowering the object.
Other common measurements
| Quantity | Instrument and useful precaution |
|---|---|
| Mass | Use a balance with a suitable range. Check its zero; tare an empty container before adding the material, or subtract the container's mass. |
| Time interval | Use a stopwatch or suitable timer. Define the starting and finishing events clearly; use repeated events when one interval is too short to time reliably by hand. |
| Temperature | Use a thermometer or temperature probe. Put the sensing part in good contact with the material as instructed and wait for a steady reading. In a liquid, avoid resting the sensor against the container. |
| Liquid volume | Use a measuring cylinder of suitable capacity and divisions. Keep it upright and read the water meniscus at eye level. |
Record what was measured
Put the quantity and unit in each table heading, such as Time for 20 oscillations / s. The entries then contain the numerical readings. Keep readings consistent with the scale or display used; do not invent extra decimal places to make a result look more precise.
Keep enough digits during a calculation to avoid unnecessary rounding, then report the result with precision justified by the measurements. Show a mean or a corrected reading separately so the original observations remain clear.
Some quantities are calculated from several measurements. For example, measuring distance and elapsed time lets you calculate average speed. The examples below apply the same measurement principles to particular quantities.
Find a method for a particular quantity
- Motion: average speed and free-fall acceleration with light gates.
- Forces and materials: mass and force, density and surface pressure.
- Turning effects: moments and the centre of gravity of a lamina.
- Energy and power: useful lifting power and electrical input energy and power.
- Temperature: temperature readings, heat capacity and latent heat.
- Waves: wavelength, period and frequency, and wave speed.
- Light: reflection angles, refraction angles and focal length; also critical angle.
- Circuits: current and charge, voltage and e.m.f., and resistance.
- Radioactivity: background correction and calibrated activity, and half-life from a decay curve.
05
Practical skills: tables, graphs and errors
Make a guess. You are not marked.
You time 10 swings of a pendulum three times and get 12.6 s, 12.4 s and 12.8 s. What value should you record for the time of 10 swings?
Show the answer
12.6 s, the mean of the three readings.
Repeated readings scatter because of reaction time. Their mean, (12.6 + 12.4 + 12.8) / 3 = 12.6 s, is the best value to record.
Paper 3, the practical paper, is worth 20% of your grade. Most of its marks reward the same few habits in every experiment. Learn them once and use them every time.
1. Record each reading to the precision of the instrument
Write every reading to the smallest division or display step of the instrument you used. A rule marked in millimetres reads to 0.1 cm, so a length that ends exactly on the 12 cm mark is 12.0 cm, not 12 cm. For a digital instrument, write exactly what the display shows.
| Instrument | Record to | Example |
|---|---|---|
| Metre rule (mm markings) | 0.1 cm | 45.0 cm |
| Digital calipers | 0.01 mm, as displayed | 12.34 mm |
| Digital stopwatch | 0.01 s, as displayed | 15.62 s |
| Digital ammeter or voltmeter | Every digit displayed | 0.38 A |
Readings from one instrument should have the same number of decimal places all the way down a column.
2. Draw the table so that it can be marked
- Heading = quantity / unit. Write l / cm or R / Ω at the top of each column. Put units only in the heading, never next to each number.
- Take at least 5 sets of readings. Spread them evenly over the widest range the apparatus allows.
- Repeat and average when readings vary, for example timing a swing or a falling object. Keep the raw readings in the table as well as the mean.
3. Give calculated values to sensible significant figures
A calculated value should have the same number of significant figures as the least precise reading you used, or one more. Keep extra digits while you calculate. Round only the final answer.
Worked example
How many significant figures?
A voltmeter reads 2.46 V (3 significant figures) and an ammeter reads 0.38 A (2 significant figures).
R = V / I = 2.46 / 0.38 = 6.4736... Ω
Write 6.5 Ω (2 s.f.) or 6.47 Ω (3 s.f.).
Writing 6.4736842 Ω claims far more precision than a 2-figure ammeter reading can give. Most mark schemes accept 2 or 3 significant figures in a final answer.
Try it step by step
Work it out in steps: Calculator value of R = V / I in ohm, then Fewest significant figures in the data, then R to that number of sf in ohm.
Show the answer
Problem 1.
Calculator value of R = V / I in ohm: R = 3.72 / 0.45 = 8.2666... ohm
Fewest significant figures in the data: 3.72 V has 3 sf and 0.45 A has 2 sf, so the data justify 2 sf
R to that number of sf in ohm: R = 8.3 ohm (2 sf)
Problem 2.
Calculator value of R = V / I in ohm: R = 5.1 / 0.235 = 21.702... ohm
Fewest significant figures in the data: 5.1 V has 2 sf and 0.235 A has 3 sf, so 2 sf
R to that number of sf in ohm: R = 22 ohm (2 sf)
Problem 3.
Calculator value of R = V / I in ohm: R = 2.84 / 0.62 = 4.5806... ohm
Fewest significant figures in the data: 2.84 V has 3 sf and 0.62 A has 2 sf, so 2 sf
R to that number of sf in ohm: R = 4.6 ohm (2 sf)
4. Plot a graph the way it is marked
- Plot the right way round. "Plot R against l" means R goes up the vertical axis and l goes along the horizontal axis.
- Label both axes with the quantity and unit, written the same way as the table heading, such as R / Ω.
- Choose an easy scale. Let each large square stand for 1, 2 or 5 units (or 10, 20, 50 and so on). Never use 3, 6 or 7: these make points hard to plot and read.
- Fill at least half the grid in both directions. The axes do not need to start at zero unless the question asks for it.
- Plot each point carefully, to within half a small square, as a small cross or a dot with a circle round it.
- Draw one thin line of best fit. Use a ruler for a straight line, or draw one smooth curve. Leave about as many points above the line as below it. Never join the dots point to point.
5. Find a gradient from a large triangle
Draw a right-angled triangle on your best-fit line that uses more than half the length of the line. Read the two corners from the line itself, not from your table, unless a data point lies exactly on the line. Write both coordinates in your working.
Worked example
Gradient of a resistance-length graph
A student measured the resistance R of different lengths l of one wire.
| l / cm | R / Ω |
|---|---|
| 20.0 | 3.1 |
| 30.0 | 4.4 |
| 40.0 | 6.1 |
| 50.0 | 7.4 |
| 60.0 | 9.0 |
| 70.0 | 10.6 |
Her best-fit straight line passes through the origin. She picks two points on the line, (20.0 cm, 3.0 Ω) and (70.0 cm, 10.5 Ω), which span the whole drawn line.
Gradient = (10.5 - 3.0) / (70.0 - 20.0)
= 7.5 / 50.0 = 0.150 Ω/cm
The gradient has a unit: the y-unit divided by the x-unit. Here it is the resistance of each centimetre of wire. A straight line through the origin shows that R is directly proportional to l.
6. Name a real source of error and a matching improvement
"Human error" or "repeat the experiment" on its own scores nothing. Name the specific problem, say how it affects the reading, and give an improvement that fixes that problem.
| Source of error | Matching improvement |
|---|---|
| Reaction time is large compared with the time of one swing. | Time 20 oscillations and divide by 20. |
| Parallax when reading a scale from an angle. | Place the eye directly in line with the mark, perpendicular to the scale. |
| The wire heats up, so its resistance rises during the experiment. | Use a small current and open the switch between readings. |
| The hot liquid loses energy to the surroundings. | Insulate the beaker and cover it with a lid. |
You will also be asked to plan an experiment of your own. Planning an experiment shows the full structure, with a worked plan.
Try it step by step
Work it out in steps: Choose the y-axis scale, then Gradient from a large triangle in cm/N, then Intercept on the length axis in cm.
Show the answer
Problem 1.
Choose the y-axis scale: Use 2 cm per large square, 0 to 12 cm: the points then fill the whole grid and every reading is easy to plot.
Gradient from a large triangle in cm/N: Draw the best-fit line and a triangle covering most of it: gradient = (12.0 - 4.0) / (5.0 - 0) = 1.6 cm/N
Intercept on the length axis in cm: The line meets the length axis at F = 0: intercept = 4.0 cm, the unstretched length
Problem 2.
Choose the y-axis scale: Use 2 cm per large square, 0 to 16 cm, which fills the grid.
Gradient from a large triangle in cm/N: Gradient = (16.0 - 6.0) / (4.0 - 0) = 2.5 cm/N
Intercept on the length axis in cm: Intercept = 6.0 cm
Problem 3.
Choose the y-axis scale: 2 cm per large square, 0 to 12 cm: the points fill most of the grid and the scale is easy to read.
Gradient from a large triangle in cm/N: Gradient = (11.0 - 3.0) / (10.0 - 0) = 0.80 cm/N
Intercept on the length axis in cm: Intercept = 3.0 cm
Check your understanding
A student measures the length of a spring with a metre rule marked in millimetres. The spring ends exactly at the 12 cm mark. How should she record the length in her table?
Show the answer
12.0 cm
The rule is marked in millimetres, which is 0.1 cm. So record every reading to 0.1 cm, even when the last digit is zero.
06 / Pure
Planning an experiment
Make a guess. You are not marked.
You test how a pendulum's length affects its period. What must you keep the same?
Show the answer
The amplitude and bob mass, so that length is the one variable that changes.
In a fair test you change the independent variable, length, measure the dependent variable, period, and keep everything else that could affect it the same.
A planning question asks you to describe an experiment you will not carry out. The marks go to a clear structure: what you change, what you measure, what you keep the same, and how the results answer the question.
The six parts of a plan
- Variables. The independent variable is what you change. The dependent variable is what you measure. Controlled variables are what you keep the same, and you say how.
- Apparatus. List it and draw a labelled diagram of the set-up.
- Procedure. Write numbered steps. Say how you measure each quantity and with which instrument.
- Range. Use at least 5 values of the independent variable over a wide range. Repeat each reading and take an average.
- Table, graph and conclusion. Give the table headings, say what to plot against what, and say what the graph shape or gradient would show.
- Risks and precautions. Name a real hazard in this experiment and the precaution that keeps the risk small.
Worked plan
How does the resistance of a wire depend on its length?
Task. You have a long piece of resistance wire, a metre rule, a cell, a switch, an ammeter, a voltmeter, connecting leads and two crocodile clips. Plan an experiment to find how the resistance R of the wire depends on its length l.
Variables. The independent variable is the length l of wire between the clips. The dependent variable is the resistance R, calculated from R = V / I. Keep the material and the diameter of the wire constant by using the same piece of wire for every length. Keep the temperature of the wire steady by using a small current and opening the switch between readings.
Apparatus and set-up. Tape the wire straight along the metre rule. Connect the cell, switch and ammeter in series with the wire, using one crocodile clip fixed at the 0.0 cm mark and one sliding clip. Connect the voltmeter across the wire between the two clips.
Procedure.
- Place the sliding clip so that l = 20.0 cm. Read l from the metre rule with your eye directly above the clip.
- Close the switch. Record the voltmeter reading V and the ammeter reading I. Open the switch straight away.
- Calculate R = V / I.
- Repeat for l = 30.0, 40.0, 50.0, 60.0 and 70.0 cm.
- Repeat the whole set of readings and find the mean R for each length.
Table. l / cm, V / V, I / A, R / Ω, and a column for the mean R / Ω.
Graph and conclusion. Plot R against l. If the points lie on a straight line through the origin, R is directly proportional to l. The gradient is the resistance of each centimetre of the wire. The gradient example uses results from this plan.
Risks and precautions. A short length of wire can draw a large current and become hot enough to burn. Keep the current small, open the switch between readings, and do not touch the wire while the switch is closed.
Accuracy. Check that both meters read zero before you start. Make sure the clips grip the bare wire firmly, and measure l between the points where the clips touch the wire.
Use the same six headings for any plan: how the period of a pendulum depends on its length, how the temperature of water changes as it cools, or how the current in a circuit depends on the voltage. Only the details change.
Check your understanding
A student plans to investigate how the resistance of a wire depends on its length. Which quantity must she keep constant?
Show the answer
The diameter of the wire
A thicker wire has a smaller resistance, so the diameter must stay the same. Using one piece of wire for every length keeps both the diameter and the material constant.
07
Scalars and vectors
Make a guess. You are not marked.
You walk 3 m east, then 4 m north. What is your displacement from where you started?
Show the answer
5 m, at an angle north of east.
Displacement is a vector: the straight line from start to finish. The two walks form a right angle, so the size is sqrt(32 + 42) = 5 m, pointing north of east.
A scalar has magnitude only. A vector has both magnitude and direction.
Magnitude means size, given by a numerical value and its unit where applicable. A speed of 4 m/s tells us how fast an object moves. A velocity of 4 m/s east also tells us its direction.
| Scalar | Related vector |
|---|---|
| Distance: total length of the path travelled. | Displacement: change in position from start to finish, including direction. |
| Speed: distance travelled per unit time. | Velocity: rate of change of displacement, including direction. |
| Mass: a property measured in kilograms. | Weight: the gravitational force on an object, measured in newtons (N). Near Earth it acts downwards. |
Time, temperature and energy are also scalars. Force and acceleration are vectors. Acceleration describes a change in velocity per unit time, so the direction of that change matters.
Compare the descriptions
Same magnitude does not mean the same vector
Two cyclists each have a speed of 4 m/s. One travels east and one west. Their speeds are equal, but their velocities differ because they point in opposite directions.
If east is chosen as positive along a straight line, the velocities can be written as +4 m/s and -4 m/s. The signs now encode the chosen directions; both magnitudes are 4 m/s.
A minus sign alone does not make a quantity a vector. A temperature of -5 °C is still a scalar. The value is below zero on the Celsius scale; it does not point in a spatial direction.
In a vector answer, state the direction as well as the magnitude, such as 6 N upwards. For signed values, say which direction you have taken as positive.
Check your understanding
Which of these describes a vector quantity?
Show the answer
A velocity of 4 m/s west.
Velocity needs both a magnitude (4 m/s) and a direction (west). A velocity of 4 m/s east would be a different vector.
When two vectors act together, use a scale drawing to find their resultant.
08 / Pure
Adding vectors with a scale drawing
Make a guess. You are not marked.
Two forces of 3 N and 4 N act on the same point in any directions. Which value could NOT be their resultant?
Show the answer
8 N, because the largest possible resultant is 7 N.
The resultant is largest, 7 N, when the forces point the same way, and smallest, 1 N, when they are opposite. Any value between is possible; 8 N is not.
The resultant is the single vector equal to the sum of the vectors being added. A scale drawing lets you find its magnitude and direction.
A vector arrow represents both magnitude and direction. Keep both unchanged when moving an arrow for the construction.
First check the directions
Suppose two forces act at the same time on one object. If they are 3 N east and 2 N east, their resultant is 5 N east. If they are 3 N east and 2 N west, their resultant is 1 N east. Adding magnitudes alone would ignore the opposing direction.
For forces that are not along the same line, construct their vector sum head to tail. Draw the second arrow starting at the head of the first, without rotating it or changing its length. The resultant points from the first arrow's tail to the second arrow's head.
Moving the arrow is a drawing method for adding the vectors. In these examples, both forces act simultaneously on the same object; the arrows do not show forces applied one after the other.
Worked construction
3 N east and 4 N north
Two simultaneous forces act on one object: 3 N east and 4 N north. Find their resultant using a scale drawing.
- Choose a scale: on paper, use 1 cm to represent 1 N.
- Draw the first vector: draw AB 3 cm long towards the east.
- Draw the second vector: from B, draw BC 4 cm long towards the north. It makes a right angle with the east direction.
- Join the endpoints: draw the resultant arrow from A to C.
- Measure: AC is 5 cm long and makes an angle of about 53° north of east.
- Convert back: 5 cm represents 5 N, so the resultant is 5 N at about 53° north of east.
Construct the scale drawing on paper to measure it. The diagram on a screen changes size with the display, so its lengths need not be physical centimetres. Use the supplied lengths and scale. The method is construction and measurement, not remembering one particular triangle.
The method also works without a right angle
Now two simultaneous forces are 4 N east and 3 N at 60° north of east. The second force is diagonal, rather than directly north.
- Use the same paper scale, 1 cm per N. Draw AB 4 cm east.
- At B, draw a light reference line continuing east. Use a protractor to measure 60° towards the north from that line.
- Along that direction draw BC 3 cm long. Join A to C, with the arrowhead at C.
- Measure AC: about 6.1 cm. Its direction is about 25° north of east.
- The resultant is therefore about 6.1 N at 25° north of east.
The answer is less than 4 + 3 = 7 N because the forces are not in the same direction. An arrow from C to A would have the correct length but the wrong direction for the resultant.
Check your understanding
Two force arrows are drawn head to tail: the first points from A to B and the second from B to C. Which arrow represents their resultant?
Show the answer
A to C
The resultant joins the tail of the first vector to the head of the second. Moving an arrow for the construction is allowed only if its length and direction stay unchanged.
Revision summary
A measurement needs a clear quantity, a sensible unit and a method suitable for the object or event. Use the links below when you need the reasoning behind a rule.
Base quantities and units
| Quantity | SI unit | Symbol |
|---|---|---|
| Length | metre | m |
| Mass | kilogram | kg |
| Time | second | s |
| Electric current | ampere | A |
| Thermodynamic temperature | kelvin | K |
| Amount of substance | mole | mol |
A quantity symbol describes the variable; a unit symbol identifies its unit. In l = 2.0 m, lowercase l means length and m means metre. Kelvin is K, without a degree sign.
Prefixes and conversions
| Prefix | Symbol | Factor |
|---|---|---|
| nano | n | 10-9 |
| micro | µ | 10-6 |
| milli | m | 10-3 |
| centi | c | 10-2 |
| deci | d | 10-1 |
| kilo | k | 103 |
| mega | M | 106 |
| giga | G | 109 |
| tera | T | 1012 |
For area, square the length conversion factor; for volume, cube it. Check the order of size before accepting a result: a hair is much thinner than a coin.
Choose an instrument
- Range: can it cover the measurement?
- Resolution: are its divisions or display steps small enough?
- Method: can it be aligned, read and used without distorting the object?
Precision concerns agreement between repeats; accuracy concerns closeness to the true or accepted value. More displayed digits alone guarantee neither.
Read, correct and calculate
- Length from a rule
- Final end reading - initial end reading. Look perpendicularly at the scale to avoid parallax.
- Stable zero offset
- Corrected reading = displayed reading - signed zero offset. Repeating alone does not remove the offset.
- Period of a repeated motion
- Mean elapsed time for N complete oscillations / N. Use the same reference point and direction for each count.
- Volume by displacement
- Final water volume - initial water volume. Fully submerge a suitable solid; avoid trapped air and lost water. Read the lower water meniscus at eye level.
Use quantity/unit headings in data tables. Record readings consistently with the instrument; preserve the original observations when averaging or correcting. The measurement-method directory links to examples for motion, forces, thermal changes, waves, optics, circuits and radioactivity.
Practical skills
- Readings: record to the instrument's precision, such as 12.0 cm on a millimetre rule.
- Tables: heading = quantity / unit; at least 5 sets of readings over a wide range.
- Calculated values: same number of significant figures as the least precise reading, or one more.
- Graphs: labelled axes with units, an easy scale (1, 2 or 5 per square, never 3, 6 or 7), points filling half the grid, one thin best-fit line.
- Gradient: a triangle using more than half the line, corners read from the line, answer with a unit.
- Errors: name a specific source of error and an improvement that fixes it.
- Plans: variables, apparatus, procedure, range, table and graph, risks and precautions. See the worked plan.
Scalars and vectors
- Scalars have magnitude only: distance, speed, mass, time, temperature and energy.
- Vectors have magnitude and direction: displacement, velocity, acceleration and force, including weight.
- A negative value alone does not make a quantity a vector. State a vector's direction or the chosen positive direction.
Pure: add two vectors graphically
- Choose and state a useful scale.
- Draw both vectors head to tail, preserving their magnitudes and directions.
- Draw the resultant from the first tail to the final head.
- Measure its length and direction; convert the length back using the scale.
Use a paper construction for physical ruler measurements. A resized screen image does not preserve centimetres.
Back to physical quantities and unitsPractise exam questions
Write your answer first, then tick the mark points it makes.
Exam-style question
A student uses a stopwatch to time 20 complete swings of a pendulum. The reading is 32.4 s. Calculate the period of the pendulum. Explain why timing 20 swings gives a more accurate period than timing one swing. [4]
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Exam-style question
State the difference between a scalar quantity and a vector quantity. Then sort these quantities into scalars and vectors: mass, velocity, distance, weight. [3]
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Exam-style question
Choose the most suitable instrument (metre rule, vernier calipers or micrometer screw gauge) to measure each of these: (a) the length of a laboratory bench, about 1.5 m; (b) the internal diameter of a test tube, about 1.6 cm; (c) the diameter of a copper wire, about 0.5 mm. Give one reason for choice (c). [3]
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Exam-style question
Two forces act on a point: 6.0 N towards the east and 8.0 N towards the north. Find the size and direction of the resultant force, either by calculation or by a scale drawing. [3]
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Exam-style question
Plan an experiment to find out how the period of a simple pendulum depends on its length. Your plan should name the variables, describe the procedure and say how you would use your results. [5]
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Quick recall
A current is 250 mA. Which conversion to amperes is correct?