IGCSE Physics Notes · Chapter 1

Measurements & Physical Quantities

How to measure length, volume and time properly, why we time many swings of a pendulum instead of one, and how vectors add up.

✓ Free chapter Syllabus 1.1 Core + Extended 10-question quiz
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Syllabus Contents 1.1 Length 1.2 Volume 1.3 Time 1.4 Multiples 1.5 Scalars & vectors 1.6 Resultants Key terms Worked examples Exam tips Experiments Quiz
You are viewing Core only. Sections 1.5 and 1.6 (scalars, vectors and resultants) are Extended content and are hidden. Switch to Extended (at the top, or in the Chapters menu on a phone) to see them.

What you need to know

Cambridge IGCSE Physics 0625 · Syllabus 1.1 Physical quantities and measurement techniques. Tick each one when you can do it. Your ticks are saved on this device.

Core

  • 1Describe how to use rulers and measuring cylinders to find a length or a volume.
  • 2Describe how to measure different time intervals using clocks and digital timers.
  • 3Find an average value for a small distance or a short time interval by measuring multiples, including the period of a pendulum.

Extended (Supplement)

  • 4Understand that a scalar has size (magnitude) only, while a vector has size and direction.
  • 5Know the scalars: distance, speed, time, mass, energy and temperature.
  • 6Know the vectors: force, weight, velocity, acceleration, momentum, electric field strength and gravitational field strength.
  • 7Find the resultant of two vectors at right angles by calculation or by scale drawing (forces or velocities only).

1.1Measuring length Core

Rulers, metre rules and tape measures, and how to avoid parallax error.

Length is the distance between two points. Its SI unit is the metre (m). For smaller or larger lengths we use millimetres, centimetres and kilometres.

Big unit → small unit: MULTIPLY Small unit → big unit: DIVIDE kmmcmmm kilometremetrecentimetremillimetre × 1000÷ 1000 × 100÷ 100 × 10÷ 10 Example: 2.5 km = 2.5 × 1000 = 2500 m · 76 mm = 76 ÷ 10 = 7.6 cm
Fig 1.1 Converting between units of length.

Choosing the right instrument

Pick the instrument that can measure the whole length in one go and has the finest scale for the job.

InstrumentGood forSmallest division
30 cm rulerLengths up to 30 cm, e.g. a pencil or a book1 mm
Metre ruleLengths up to 1 m, e.g. a pendulum string or a bench1 mm
Tape measureSeveral metres, e.g. a room or a long jump1 mm or 1 cm

How to use a ruler correctly

  1. Place the ruler flat against the object, as close to it as possible.
  2. Don't start from the very end of the ruler. The end is often worn or has a gap before the zero mark. Line the object up with a clear mark (for example 1.0 cm) and subtract that reading at the end.
  3. Put your eye directly above the mark you are reading, looking straight down. Looking from an angle gives a wrong reading. This is called parallax error.
  4. Record to the nearest millimetre, with the unit: for example 7.6 cm or 76 mm. A ruler can't give you 7.63 cm.
✗ Worn end: where is zero? 12 The 0 mark is chipped away, so a reading taken from the end would be wrong. ✓ Start at a clear mark 012 start = 1.0 cm, end = 2.3 cm length = 2.3 − 1.0 = 1.3 cm (1 small division = 1 mm)
Fig 1.2 Never trust a worn ruler end. Line the object up with a clear mark and subtract.
012345678910 cm ✓ Eye straight above ✗ At an angle: reads 8.3 cm start: 1.0 cm end: 8.6 cm Length = 8.6 cm − 1.0 cm = 7.6 cm
Fig 1.3 Reading straight down gives 8.6 cm. Reading from an angle gives a false 8.3 cm: that's parallax error.
Definition

Parallax error is the error you get when you read a scale from an angle instead of looking straight at it. Avoid it by putting your eye directly in line with (perpendicular to) the scale mark.

Curved lengths

To measure a curved line, like a route on a map or the edge of a coin, lay a piece of thread along the curve. Mark the thread at the end, straighten it, and measure it with a ruler.

1. Lay thread along the curve startmark straighten 2. Measure the straight thread 0510 cm length of the curved road = 10.0 cm
Fig 1.4 Measuring a curved line with a thread.
Real life
  • A tailor measures your waist with a flexible tape measure, which bends around curves just like the thread in Fig 1.4.
  • Map apps work out the length of a winding road the same way: they add up lots of tiny straight pieces along the curve.
  • Builders say "measure twice, cut once". A parallax or zero error on one measurement can waste a whole sheet of wood.

1.2Measuring volume Core

Measuring cylinders, the meniscus, and finding the volume of solids.

Volume is the amount of space something takes up. Its SI unit is the cubic metre (m3), but in the lab we usually use cubic centimetres (cm3).

Units to remember
  • 1 cm3 = 1 ml (millilitre)
  • 1 litre = 1000 cm3
  • 1 m3 = 100 cm × 100 cm × 100 cm = 1 000 000 cm3 = 106 cm3

Volume of a liquid: the measuring cylinder

  1. Stand the measuring cylinder on a flat, level bench. Don't hold it in your hand.
  2. Water curves down slightly in the middle where it meets the glass. This curve is the meniscus.
  3. Bend down so your eye is level with the liquid surface, and read the scale at the bottom of the meniscus.
  4. Work out what one small division is worth before reading (see Fig 1.7).
  5. Choose a cylinder that fits the volume. For about 30 cm3, a 50 cm3 cylinder gives a finer reading than a 500 cm3 one.
404550 cm³ eye level meniscus (curved surface) Reading = 46 cm³
Fig 1.5 Read the bottom of the meniscus with your eye level with it.
reads too HIGH correct reads too LOW the sight line crosses the scale here
Fig 1.6 Eye above or below the surface gives a parallax error.
4050 1 division level here 10 cm³ ÷ 5 divisions = 2 cm³ per division → reading = 40 + 3 × 2 = 46 cm³
Fig 1.7 Always work out the value of one small division before you read a scale.

Volume of a regular solid

Measure its sides with a ruler and use a formula. For a cuboid, V = length × width × height. For a cylinder, V = πr2h, where r is the radius and h the height.

length = 6.0 cm height = 4.0 cm width = 3.0 cm V = l × w × h = 6.0 × 3.0 × 4.0 = 72 cm³
Fig 1.8 Volume of a cuboid from three ruler measurements.

Volume of an irregular solid: the displacement method

A stone or a key has no simple shape, so we measure the water it pushes aside (displaces). This is the displacement method.

  1. Half-fill a measuring cylinder with water and read the volume, V1.
  2. Tie the object to a thread and lower it gently until it is completely under water. Don't drop it in, or water may splash out.
  3. Read the new volume, V2.
  4. Volume of the object = V2 − V1
2030405060 Before V₁ = 35 cm³ lower the stone in gently 2030405060 After V₂ = 52 cm³ rise = 17 cm³ = volume of stone
Fig 1.9 The water rises by exactly the volume of the stone: V = 52 − 35 = 17 cm3.

If the object is too big for a measuring cylinder, use a displacement (overflow) can. Fill the can until water just drips from the spout, lower the object in, and collect the overflowing water in a measuring cylinder. The volume collected is the volume of the object.

overflow (displacement) can object lowered on a thread displaced water collected volume collected = volume of object Start with water just dripping from the spout.
Fig 1.10 A displacement (overflow) can for objects too big for a measuring cylinder.
Try it in the 3D lab

Find the volume and density of irregular objects yourself in our Displacement Method simulator. You'll use this again in Chapter 3 to find density.

Real life
  • The measuring cup that comes with cough syrup is marked in ml. Hold it at eye level to read 5 ml correctly, because the meniscus matters for medicine too.
  • At a petrol pump the meter shows the volume of fuel in litres (1 litre = 1000 cm³).
  • Water tanks on rooftops are sold by volume, for example 1000 litres = 1 m³.

1.3Measuring time Core

Clocks, stopwatches, light gates, and the problem of reaction time.

The SI unit of time is the second (s). Remember 1 minute = 60 s and 1 hour = 3600 s.

Which timer should you use?

Clock minutes – hours Analogue stopwatch seconds – minutes 00:12.34 Digital stopwatch seconds – minutes (shows 0.01 s) Light gate + timer fractions of a second longer time intervals shorter time intervals
Fig 1.11 Choose the timer to match how long the interval is.

Reaction time

When you use a stopwatch, you need time to react: to see the event and press the button. Your reaction time is usually about 0.2–0.3 s. So although a digital stopwatch shows 0.01 s, your timing is much less precise than that.

00.10.20.30.40.5time / s reaction time ≈ 0.25 s event happens (e.g. ball released) you press the button (stopwatch starts late) The same delay happens again when you stop the watch.
Fig 1.12 Reaction time: the gap between an event and your finger pressing the button.

This is why light gates are used for very short times. The timer starts and stops automatically when an object breaks the beam of light, so there is no human reaction time.

card moving light gate light beam across the track 0.080 s electronic timer starts when the card cuts the beam, stops when it leaves
Fig 1.13 A light gate times a trolley automatically, with no reaction time.
Reading a digital stopwatch

Convert minutes to seconds before you calculate anything.

01:23.45 minutessecondshundredths 01:23.45 = 60 s + 23.45 s = 83.45 s
Fig 1.14 Reading a digital stopwatch display.
Real life
  • In athletics, races are timed electronically to 0.01 s. A starting gun triggers the clock, and a camera at the finish line records each runner, so no one's reaction time affects the result.
  • Cricket's third umpire uses high-speed cameras that take hundreds of pictures per second to judge run-outs that the eye cannot see.
  • Your phone's stopwatch shows hundredths of a second, but your thumb is only accurate to about 0.2 s, so the last digit is mostly guesswork.

1.4Measuring multiples & finding averages Core

How to measure things that are too small or too quick to measure once, including the period of a pendulum.

Some things are too small or too quick to measure accurately on their own, like the thickness of one sheet of paper or the time of one pendulum swing. The trick is to measure multiples: measure many of them together, then divide.

Small distances

4.4cm 400 sheets one sheet = 4.4 cm ÷ 400 = 0.011 cm = 0.11 mm
Fig 1.15 Thickness of one sheet of paper.
20 turns = 1.2 cm thin wire wound tightly on a pencil diameter = 1.2 cm ÷ 20 = 0.06 cm = 0.6 mm
Fig 1.16 Diameter of a thin wire.

The period of a pendulum

Definition

One oscillation is one complete swing: from one side, across to the other side, and back again (A → B → A). The period (T) is the time taken for one complete oscillation.

AB AB AB AB AB 1 starts at A 2 passes centre → 3 reaches B 4 passes centre ← 5 back at A A → B is only HALF an oscillation A → B → A = ONE complete oscillation · time taken = period T
Fig 1.17 One complete oscillation, step by step. The green line under each pendulum is the fiducial marker at the centre of the swing.
A B fiducial marker (start and stop counting here) 1 oscillation = A → B → A length of pendulum
Fig 1.18 Pendulum set-up. Time 20 complete oscillations, counting each time the bob passes the fiducial marker in the same direction.

Method

  1. Pull the bob a small distance to one side and release it gently. Don't push it.
  2. Place a fiducial marker (a pointer or a line) at the centre of the swing. The bob moves fastest here, so you can see exactly when it passes.
  3. Let it swing a few times. Start the stopwatch as the bob passes the marker and count "zero".
  4. Count each time it passes the marker in the same direction. Stop the stopwatch at "20".
  5. Calculate the period: T = time for 20 oscillations20
  6. Repeat the timing two or three times and use the average.

Why time 20 oscillations instead of 1?

Your reaction time adds roughly the same error, about 0.2 s, whether you time 1 swing or 20 swings. Dividing by 20 also divides the error by 20, so the period is much more accurate.

Timing 1 swing reading ≈ 1.5 s 0.2 s 13% Timing 20 swings reading ≈ 30 s 0.7% Each bar is the whole reading. Red is the 0.2 s reaction-time error. Timing more oscillations makes the same error a much smaller part of the result.
Fig 1.19 Why measuring multiples gives a more accurate period.

Repeating and averaging

Repeating a measurement and finding the mean (average) reduces the effect of small random errors:

mean = sum of the readingsnumber of readings

If one reading is clearly very different from the others, it is an anomalous result. Repeat that measurement if you can, and leave the anomaly out when you calculate the mean.

1516171819t / s 15.1, 15.2, 15.3, 15.4 mean = 15.25 s 18.9 s: anomalous leave it out of the mean Five repeat readings of the time for 10 oscillations
Fig 1.20 Spotting an anomalous result before you average.
Good to know for the practical paper

A zero error happens when an instrument doesn't read zero when it should, like a newton meter showing 0.2 N with nothing hanging on it. Every reading is then wrong by the same amount, so check the zero before you start and correct for it.

Try it in the 3D lab

Time 20 oscillations, change the length and calculate g in our Simple Pendulum simulator.

Real life
  • Old grandfather clocks keep time with a pendulum. Clockmakers adjust the length of the pendulum until its period is exactly right.
  • Shops and banks can count a bundle of identical coins or notes by weighing the whole bundle and dividing by the mass of one. That is measuring multiples with mass instead of length.
  • Doctors count your pulse for 15 or 30 seconds and multiply up, rather than timing a single beat. That is measuring multiples in reverse.

1.5Scalars and vectors Extended

Quantities with size only, and quantities with size and direction.

Definitions

A scalar quantity has magnitude (size) only.
A vector quantity has magnitude and direction.

SCALAR size only: a number + a unit 5 kg 30 °C 12 s 400 J VECTOR size + direction: drawn as an arrow force = 5 N to the right velocity = 3 m/s north
Fig 1.21 A scalar is just a size. A vector needs a size and a direction, so we draw it as an arrow.

You must know these lists for the exam:

Scalars (size only)Vectors (size + direction)
distance
speed
time
mass
energy
temperature		force
weight
velocity
acceleration
momentum
electric field strength
gravitational field strength

Displacement (distance in a stated direction) is also a vector, and you'll meet it in Chapter 2.

A quick test

Ask whether the quantity makes sense with a direction attached. "A force of 20 N downwards" makes sense, so force is a vector. "A temperature of 30 °C to the north" makes no sense, so temperature is a scalar.

Distance vs displacement, speed vs velocity

Distance is how far you actually travel along your path (scalar). Displacement is the straight-line distance from start to finish, in a stated direction (vector).

Walk 3 m east, then 4 m north start finish 3 m 4 m 5 m distance = 3 + 4 = 7 m (scalar: the path you walked) displacement = 5 m (vector: straight line, start → finish, at 53° north of east) To the shop and back home shop 50 m there 50 m back distance = 100 m displacement = 0 (you end where you started)
Fig 1.22 Distance follows your path. Displacement is the straight arrow from start to finish.

In the same way, a car's speed might be 20 m/s, while its velocity is 20 m/s due north.

Watch out

Mass is a scalar, but weight is a vector. Weight is a force, and it always acts downwards, towards the centre of the Earth.

Drawing vectors

A vector is drawn as an arrow. The length of the arrow shows the magnitude (drawn to a scale) and the arrowhead shows the direction. Along a straight line, we often use + for one direction and − for the other.

Scale: 1 cm = 1 N (here 1 cm is drawn as 40 px) 2 N right → 2 cm long → written +2 N 4 N right → 4 cm long → written +4 N 3 N left → 3 cm long → written −3 N
Fig 1.23 Arrow length shows size; arrowhead shows direction. Twice the force means twice the length.
Real life
  • A taxi meter charges you for distance (the path driven), not displacement. A round trip back to your door still costs money even though your displacement is zero.
  • Weather forecasts give wind as a vector, for example "25 km/h from the north-west", because the direction matters as much as the speed.

1.6Adding vectors: the resultant Extended

Vectors in the same direction, in opposite directions, and at right angles.

Definition

The resultant of two or more vectors is the single vector that has the same effect as all of them acting together.

Vectors in the same direction (parallel): add

Join the arrows tip-to-tail. The resultant goes from the start of the first arrow to the tip of the last, so just add the sizes.

30 N 20 N Both push to the right tip-to-tail: 30 N20 N resultant: 50 N 30 N + 20 N = 50 N to the right
Fig 1.24 Same direction: add the sizes. The resultant points the same way.

Vectors in opposite directions (anti-parallel): subtract

Take one direction as positive. Subtract the smaller from the larger, and the resultant points in the direction of the larger vector.

200 N 150 N (+200) + (−150) = +50 N resultant = 50 N to the right Unbalanced
Fig 1.25 Opposite directions: subtract. The bigger force wins.
100 N 100 N (+100) + (−100) = 0 resultant = 0 forces are balanced Balanced
Fig 1.26 Equal and opposite: the resultant is zero.

Two vectors at right angles

Now the arrows form a right-angled triangle, so the resultant is not the simple sum. A 12 N force east and a 5 N force north give a resultant of 13 N, not 17 N.

12 N east 5 N north resultant 13 N θ Triangle (tip-to-tail) method
Fig 1.27a Draw the second arrow from the tip of the first. The resultant goes from the start to the finish.
12 N east 5 N north resultant 13 N Parallelogram method
Fig 1.27b Draw both arrows from one point and complete the rectangle. The diagonal is the resultant. Both methods give the same answer.

Method 1: by calculation

Use Pythagoras' theorem for the size of the resultant, and tan for its direction:

R = √( F12 + F22 )      tan θ = oppositeadjacent

For the forces in Fig 1.27: R = √(122 + 52) = √169 = 13 N, and tan θ = 5 ÷ 12, so θ = 22.6°. The resultant is 13 N at 22.6° north of east.

Method 2: by scale drawing

  1. Choose a sensible scale and write it down, e.g. 1 cm = 1 N. Make the diagram as large as the space allows.
  2. Draw the first vector as an arrow of the right length and direction.
  3. From the tip of the first arrow, draw the second at 90° using a protractor or set square.
  4. Draw the resultant from the start of the first to the tip of the second.
  5. Measure its length with a ruler and convert using your scale. Measure the angle with a protractor.
8 cm = 8 N 6 cm 37° 10 cm 1. Draw 8 N eastto scale (1 cm = 1 N) 2. From its tip,draw 6 N north at 90° 3. Join startto finish 4. Measureruler + protractor Resultant = 10 cm → 10 N at 37° north of east
Fig 1.28 Finding a resultant by scale drawing, step by step. Check: √(8² + 6²) = 10 N and tan θ = 6/8 gives θ = 37°.

Velocities add the same way

An aircraft pointing north at 120 m/s through air that moves east at 50 m/s actually travels at 130 m/s, slightly east of north (worked example 6).

θ aircraft 120 m/s N wind 50 m/s E resultant 130 m/s tan θ = 50 ÷ 120 θ = 22.6° east of north N
Fig 1.29 Adding two velocities at right angles.
Always give the direction

A resultant is a vector, so "13 N" on its own is incomplete. Write "13 N at 22.6° north of east" or "13 N at 22.6° to the 12 N force".

Real life
  • In a tug of war, the team that wins is the one on the side of the resultant force. If the teams pull equally, the resultant is zero and the rope does not move.
  • A boat crossing a flowing river is pushed sideways by the current, so it lands downstream. The boat's velocity and the river's velocity add as vectors at right angles.
  • Pilots point an aircraft slightly into a crosswind so that the resultant velocity takes them straight to the runway.

Key terms & formulae

Learn these word for word. Definitions earn easy marks.

LengthThe distance between two points. SI unit: metre (m).
VolumeThe space taken up by an object. SI unit: m3. 1 cm3 = 1 ml.
MeniscusThe curved surface of a liquid in a tube. Read the bottom of it for water.
Parallax errorAn error from reading a scale at an angle. Avoid it by looking straight at the mark.
Reaction timeThe delay between seeing an event and pressing the stopwatch, about 0.2–0.3 s.
OscillationOne complete swing of a pendulum, from one side to the other and back.
Period (T)The time taken for one complete oscillation.
Fiducial markerA fixed reference mark at the centre of the swing, used for counting oscillations.
Anomalous resultA reading that doesn't fit the pattern of the others. Leave it out of the average.
Scalar ExtA quantity with magnitude (size) only.
Vector ExtA quantity with magnitude and direction.
Resultant ExtThe single vector with the same effect as two or more vectors acting together.

Formulae

length = end reading − start reading
Using a ruler
V = V2 − V1
Displacement method for an irregular solid
one = total ÷ number
Measuring multiples (sheets, turns of wire)
T = time for n oscillations ÷ n
Period of a pendulum
mean = sum ÷ number of readings
Averaging repeat readings
R = √(F12 + F22)
Resultant of two vectors at 90° Ext
tan θ = opposite ÷ adjacent
Direction of the resultant Ext

Worked examples

Try each one yourself first, then open the solution.

1A student measures a leaf. One end of the leaf is at the 2.3 cm mark of a ruler and the other end is at the 11.8 cm mark. What is the length of the leaf? Coresolution
IdeaThe leaf doesn't start at zero, so subtract the start reading from the end reading.
Workinglength = 11.8 cm − 2.3 cm = 9.5 cm
Length = 9.5 cm
2The water in a measuring cylinder reads 35 cm3. A stone is lowered in on a thread and the reading rises to 52 cm3. Find the volume of the stone in cm3 and in m3. Coresolution
IdeaThe rise in water level equals the volume of the stone.
WorkingV = V2 − V1 = 52 − 35 = 17 cm3
Convert1 m3 = 106 cm3, so 17 cm3 = 17 ÷ 1 000 000 = 1.7 × 10−5 m3
17 cm3 = 1.7 × 10−5 m3
3A student times 20 oscillations of a pendulum three times and gets 32.4 s, 32.8 s and 32.6 s. Calculate the period of the pendulum. Why did the student time 20 oscillations rather than one? Coresolution
Mean(32.4 + 32.8 + 32.6) ÷ 3 = 97.8 ÷ 3 = 32.6 s for 20 oscillations
PeriodT = 32.6 ÷ 20 = 1.63 s
Why 20?The reaction-time error (about 0.2 s) is the same whatever you time. Over 32.6 s it is a much smaller fraction of the reading than over one swing of 1.63 s, so the period is more accurate.
T = 1.63 s
425 turns of thin copper wire are wound tightly side by side around a pencil. The coil is 1.5 cm long. Find the diameter of the wire in mm. Coresolution
Idea25 diameters fit into 1.5 cm, so divide by 25.
Workingdiameter = 1.5 cm ÷ 25 = 0.06 cm = 0.6 mm
Diameter = 0.6 mm
5Two forces act on a box: 9.0 N to the east and 12.0 N to the north. Calculate the resultant force. Extendedsolution
SizeR = √(9.02 + 12.02) = √(81 + 144) = √225 = 15 N
Directiontan θ = 12.0 ÷ 9.0 = 1.33, so θ = 53° (measured from east towards north)
15 N at 53° north of east
6An aircraft points due north and flies at 120 m/s through the air. A wind blows due east at 50 m/s. Find the aircraft's resultant velocity. Extendedsolution
Sizev = √(1202 + 502) = √(14 400 + 2500) = √16 900 = 130 m/s
Directiontan θ = 50 ÷ 120 = 0.417, so θ = 22.6° (measured from north towards east)
130 m/s at 22.6° east of north

Exam tips & common mistakes

Where students most often lose marks on this topic.

Common mistakes
  • Forgetting to divide by the number of oscillations. The time for 20 swings is not the period.
  • Counting one oscillation as A → B. A full oscillation is A → B → A.
  • Reading the top of the meniscus, or reading with the eye above or below the surface.
  • Measuring from the worn end of a ruler instead of subtracting a start reading.
  • Calling mass a vector. Mass is a scalar; weight is the vector.
  • Adding perpendicular vectors as if they were in a line: 12 N + 5 N at 90° is 13 N, not 17 N.
  • Giving a resultant without its direction.
How to score full marks on "describe how to measure…"
  • Name the instrument (metre rule, measuring cylinder, stopwatch).
  • Say exactly what you measure, and the calculation if there is one (e.g. V2 − V1, or divide by 20).
  • Give a way to improve accuracy: eye level / perpendicular to the scale, measure multiples, or use a fiducial marker.
  • Finish with repeat and calculate an average.
Numbers and units
  • Always write the unit. A number on its own often scores zero.
  • Record readings to the precision of the instrument: to the nearest mm on a ruler, and to 0.01 s as shown on a digital stopwatch.
  • Give calculated answers to the same number of significant figures as the data, usually 2 or 3.
  • In tables, write column headings as quantity / unit, e.g. "time for 20 oscillations / s".

Try it in the 3D lab

The IGCSE practicals for this chapter. Ready simulators open straight away.

Chapter 1 quiz

10 exam-style multiple-choice questions. Choose an answer for each, then check your score.