What you'll learn
Measurement and physical quantities is the opening topic of CXC CSEC Physics and the one that underpins every practical question on the paper. Physics describes the world in quantities that can be measured, and a measurement is meaningless without a unit, an appropriate instrument and an honest account of its uncertainty. This topic is also where the marks in the School Based Assessment are won or lost, because it governs how you record data, plot graphs and draw conclusions. By the end of this guide you should be able to distinguish base from derived quantities and scalars from vectors, use SI units and prefixes confidently, select the right instrument for a measurement and state its resolution, use vernier callipers and micrometers, work with significant figures and standard form, identify sources of error, and plot and interpret a straight-line graph including finding its gradient and intercept.
Key terms and definitions
Physical quantity — a property that can be measured, consisting of a numerical value and a unit
Base quantity — one of the seven fundamental quantities from which all others are derived
Derived quantity — a quantity obtained by combining base quantities, such as speed or density
SI units — the International System of Units, the agreed standard for scientific measurement
Scalar — a quantity with magnitude only
Vector — a quantity with both magnitude and direction
Resolution — the smallest change an instrument can detect
Accuracy — how close a measurement is to the true value
Precision — how close repeated measurements are to one another
Systematic error — an error that shifts every reading in the same direction, often from a faulty instrument
Random error — an error that varies unpredictably between readings
Zero error — a systematic error where an instrument does not read zero when it should
Parallax error — an error caused by viewing a scale from the wrong angle
Core concepts
Base and derived quantities
There are seven base quantities in the SI system, and five matter at CSEC level: length measured in metres, mass in kilograms, time in seconds, electric current in amperes, and temperature in kelvin. The remaining two are amount of substance in moles and luminous intensity in candelas.
Every other quantity is derived by combining these. Area is length multiplied by length, giving square metres. Volume gives cubic metres. Speed is length divided by time, giving metres per second. Density is mass divided by volume, giving kilograms per cubic metre. Force is mass multiplied by acceleration, giving newtons, which are equivalent to kilogram metres per second squared.
Being able to express a derived unit in base units is examinable, and the method is simply to substitute the base units into the defining equation.
Scalars and vectors
A scalar has magnitude only: distance, speed, mass, time, energy, work, power, temperature, volume and density.
A vector has magnitude and direction: displacement, velocity, acceleration, force, weight and momentum.
The distinction has practical consequences. A person who walks 3 kilometres north and then 3 kilometres south has travelled a distance of 6 kilometres but has a displacement of zero. Scalars are added arithmetically, while vectors must be added taking direction into account.
For two vectors acting along the same line, add them if they act in the same direction and subtract if they oppose. For two vectors at right angles, the resultant is found using Pythagoras' theorem, and its direction from trigonometry.
SI prefixes
Prefixes express very large and very small quantities conveniently, and converting between them is tested constantly.
The prefixes needed are: giga meaning 10 to the power 9, mega 10 to the power 6, kilo 10 to the power 3, centi 10 to the power minus 2, milli 10 to the power minus 3, micro 10 to the power minus 6, and nano 10 to the power minus 9.
Two conversions cause most errors. One square metre is 10,000 square centimetres, not 100, because the conversion factor of 100 must itself be squared. One cubic metre is 1,000,000 cubic centimetres, because the factor is cubed. Candidates routinely convert areas and volumes as though they were lengths.
Choosing and using instruments
The instrument must suit both the size of the quantity and the precision required.
A metre rule measures lengths from a few centimetres to a metre, with a resolution of 1 millimetre.
Vernier callipers measure small lengths, internal and external diameters, and depths, with a resolution of 0.01 centimetre, which is 0.1 millimetre.
A micrometer screw gauge measures very small thicknesses and diameters, such as a wire, with a resolution of 0.01 millimetre.
A measuring cylinder measures liquid volume, read at the bottom of the meniscus with the eye level to avoid parallax.
A stopwatch measures time, though its resolution of 0.01 second is far better than human reaction time, which introduces an uncertainty of about 0.2 seconds.
A balance measures mass, and a spring balance or newtonmeter measures force and therefore weight.
Reading a vernier and a micrometer
For vernier callipers, read the main scale immediately before the zero of the vernier scale to get the whole and first decimal figure. Then find the vernier division that lines up exactly with a main scale division, and that division number gives the second decimal place. Add the two readings.
For a micrometer, read the main scale on the sleeve to the nearest half millimetre, then read the thimble scale division aligned with the horizontal line, multiply it by 0.01 millimetre and add.
Before using either, close the jaws fully and check for a zero error. If the reading is not zero when closed, every measurement carries that error, and it must be subtracted if positive or added if negative.
Accuracy, precision and errors
Accuracy concerns closeness to the true value; precision concerns closeness of repeated readings to each other. A set of readings can be precise but inaccurate, which is exactly what a systematic error produces.
Systematic errors shift every reading in the same direction and are not reduced by repeating. Zero errors, an incorrectly calibrated instrument, and consistently viewing a scale from the same wrong angle are all systematic. They are eliminated by checking and correcting the instrument.
Random errors vary unpredictably between readings, arising from reaction time, fluctuating conditions or judgement of a scale division. They are reduced by taking repeated readings and calculating a mean.
Techniques that improve measurement are examinable in their own right. Timing twenty oscillations and dividing by twenty reduces the effect of reaction time. Measuring the thickness of one hundred sheets of paper and dividing gives a better value than measuring one. Viewing a scale perpendicular to it, or using a mirror behind the pointer, eliminates parallax.
Significant figures and standard form
A calculated answer should be given to the same number of significant figures as the least precise measurement used in the calculation. Giving eight figures from a calculator suggests a precision the data does not support, and marks are deducted for it.
Standard form writes a number as a value between 1 and 10 multiplied by a power of ten. The speed of light, 300,000,000 metres per second, is written as 3 times 10 to the power 8 metres per second.
Graphs
Graph work carries substantial marks in both the written paper and the School Based Assessment, and the conventions are strict.
Plot the independent variable, the one you chose, on the horizontal axis, and the dependent variable, the one you measured, on the vertical axis.
Label each axis with the quantity and its unit, separated by a solidus, such as time in seconds.
Choose a scale that uses at least half the grid in both directions, and use sensible divisions of 1, 2, 5 or 10 units per square. Awkward scales such as 3 units per square cost marks and cause plotting errors.
Plot points as small crosses or encircled dots, and draw a single straight line of best fit with roughly equal numbers of points either side. Do not join the points dot to dot.
The gradient of a straight line is the change in the vertical value divided by the change in the horizontal value. Use a large triangle covering at least half the line, and read the coordinates of two points that lie on the line rather than two plotted data points. The gradient carries units, obtained by dividing the vertical unit by the horizontal unit.
The intercept is the value where the line crosses the vertical axis, and it can only be read directly if the horizontal axis starts at zero.
Worked examples
Example 1: Converting units correctly (3 marks)
A rectangular block measures 2.0 metres by 1.5 metres. Express its area in square centimetres.
The area is 2.0 × 1.5 = 3.0 square metres.
To convert to square centimetres, note that 1 metre is 100 centimetres, so 1 square metre is 100 × 100 = 10,000 square centimetres.
The area is therefore 3.0 × 10,000 = 30,000 square centimetres, or 3.0 × 10 to the power 4 square centimetres in standard form. Note that multiplying by 100 rather than 10,000 is the standard error here.
Example 2: Correcting a zero error (3 marks)
A micrometer reads 0.03 millimetres when its jaws are fully closed. It then gives a reading of 2.47 millimetres for the diameter of a wire. State the true diameter and the type of error.
The instrument reads 0.03 millimetres too high when it should read zero, so every measurement it gives is 0.03 millimetres too large.
The true diameter is 2.47 − 0.03 = 2.44 millimetres.
This is a systematic error, because it affects every reading by the same amount and in the same direction. Repeating the measurement would not reduce it; only correcting or recalibrating the instrument would.
Example 3: Finding a gradient (4 marks)
A straight-line graph of distance against time passes through the points (2.0 seconds, 6.0 metres) and (8.0 seconds, 27.0 metres). Calculate the gradient and state what it represents.
The gradient is the change in the vertical value divided by the change in the horizontal value.
The change in distance is 27.0 − 6.0 = 21.0 metres. The change in time is 8.0 − 2.0 = 6.0 seconds.
The gradient is 21.0 ÷ 6.0 = 3.5, and the units are metres per second.
Since the graph plots distance against time, the gradient represents the speed of the object, which is 3.5 metres per second.
Common mistakes and how to avoid them
The most frequent error is converting areas and volumes as though they were lengths. The conversion factor must be squared for area and cubed for volume.
Students often omit units from an answer or from a graph axis. A physical quantity without a unit is incomplete and loses the mark.
Another routine slip is confusing accuracy with precision. Precise readings cluster together; accurate readings sit near the true value, and a systematic error produces the first without the second.
Many candidates calculate a gradient from two plotted data points rather than from two points on the line of best fit. The line, not the data, gives the gradient.
Finally, candidates frequently give answers to the full calculator display. Match the significant figures to the least precise measurement in the question.
Exam technique for "Measurement and Physical Quantities"
Write the unit at every stage of a calculation, not just at the end. It catches conversion errors early and secures the unit mark.
When a question names an instrument, state its resolution if asked, and check whether a zero error has been mentioned. A stated zero error is always there to be applied.
For graph questions, spend the first minute choosing the scale. A poor scale costs plotting marks and makes the gradient harder to read accurately.
Use a large gradient triangle, at least half the length of the line, and mark it clearly on the graph. Examiners award marks for the construction as well as the value.
When asked how to improve an experiment, give a specific technique and say which error it reduces — repeating readings for random error, checking calibration for systematic error, timing multiple oscillations for reaction time.
Quick revision summary
Physical quantities need a value and a unit. The base quantities are length in metres, mass in kilograms, time in seconds, current in amperes and temperature in kelvin, with all others derived from them. Scalars have magnitude only while vectors also have direction, so displacement, velocity, acceleration and force must be combined with direction taken into account. SI prefixes run from giga to nano, and area conversions square the factor while volume conversions cube it. Metre rules resolve to 1 millimetre, vernier callipers to 0.01 centimetre and micrometers to 0.01 millimetre, and each must be checked for zero error before use. Systematic errors shift every reading the same way and are not reduced by repeating; random errors vary and are reduced by taking a mean. Answers should match the significant figures of the least precise measurement. Graphs need the independent variable on the horizontal axis, labelled axes with units, a scale using at least half the grid, a line of best fit rather than joined points, and a gradient found from a large triangle using points on the line.