Measurements and uncertainties

5 hours

1.1 – Measurements in physics

Essential idea: Since 1948, the Système International d’Unités (SI) has been used as the preferred language of science and technology across the globe and reflects current best measurement practice.
Nature of science: (1) Common terminology: Since the 18th century, scientists have sought to establish common systems of measurements to facilitate international collaboration across science disciplines and ensure replication and comparability of experimental findings. (2) Improvement in instrumentation: An improvement in apparatus and instrumentation, such as using the transition of cesium-133 atoms for atomic clocks, has led to more refined definitions of standard units. (3) Certainty: Although scientists are perceived as working towards finding “exact” answers, the unavoidable uncertainty in any measurement always exists.
• Fundamental and derived SI units The international mass The units
• Scientific notation and metric multipliers Practice
• Significant figures Significant figure calculator Sig fig tutorials
• Orders of magnitude Scale of the universe Practice
• Estimation Metric to English calculator
Applications and skills:
• Using SI units in the correct format for all required measurements, final answers to calculations and presentation of raw and processed data
• Using scientific notation and metric multipliers
• Quoting and comparing ratios, values and approximations to the nearest order of magnitude
• Estimating quantities to an appropriate number of significant figures
• SI unit usage and information can be found at the website of Bureau International des Poids et Mesures
• Students will not need to know the definition of SI units except where explicitly stated in the relevant topics in this guide
• Candela is not a required SI unit for this course
• Guidance on any use of non-SI units such as eV, MeV c-2, Ly and pc will be provided in the relevant topics in this guide
• Scientific collaboration is able to be truly global without the restrictions of national borders or language due to the agreed standards for data representation
Theory of knowledge:
• What has influenced the common language used in science? To what extent does having a common standard approach to measurement facilitate the sharing of knowledge in physics?
• This topic is able to be integrated into any topic taught at the start of the course and is important to all topics
• Students studying more than one group 4 subject will be able to use these skills across all subjects
• See Mathematical studies SL sub-topics 1.2–1.4
Aim 2 and 3: this is an essential area of knowledge that allows scientists to collaborate across the globe
Aim 4 and 5: a common approach to expressing results of analysis, evaluation and synthesis of scientific information enables greater sharing and collaboration

1.2 – Uncertainties and errors

Essential idea: Scientists aim towards designing experiments that can give a “true value” from their measurements, but due to the limited precision in measuring devices, they often quote their results with some form of uncertainty.
Nature of science:Uncertainties: “All scientific knowledge is uncertain… if you have made up your mind already, you might not solve it. When the scientist tells you he does not know the answer, he is an ignorant man. When he tells you he has a hunch about how it is going to work, he is uncertain about it. When he is pretty sure of how it is going to work, and he tells you, ‘This is the way it’s going to work, I’ll bet,’ he still is in some doubt. And it is of paramount importance, in order to make progress, that we recognize this ignorance and this doubt. Because we have the doubt, we then propose looking in new directions for new ideas.” Feynman, Richard P. 1998. The Meaning of It All: Thoughts of a Citizen-Scientist. Reading, Massachusetts, USA. Perseus. P 13.
• Random and systematic errors Examples and cures
• Absolute, fractional and percentage uncertainties Analog and digital uncertainites
• Error bars Using Excel
• Uncertainty of gradient and intercepts Concept Concept
Applications and skills:
• Explaining how random and systematic errors can be identified and reduced
• Collecting data that include absolute and/or fractional uncertainties and stating these as an uncertainty range (expressed as: best estimate ± uncertainty range)
• Propagating uncertainties through calculations involving addition, subtraction, multiplication, division and raising to a power
• Determining the uncertainty in gradients and intercepts
• Analysis of uncertainties will not be expected for trigonometric or logarithmic functions in examinations
• Further guidance on how uncertainties, error bars and lines of best fit are used in examinations can be found in the Teacher support material
Data Booklet reference:
• If y = a ± b then ∆y = ∆a + ∆b
• If y = ab / c then ∆y / y = ∆a / a + ∆b / b + ∆c / c
• If y = an then ∆y / y = | na / a |
• The a, b and c represent variables in formulas. The ∆ stands for "the uncertainty in". The n represents a rational power. Note that uncertainties never subtract - they always add.
Theory of knowledge:
• “One aim of the physical sciences has been to give an exact picture of the material world. One achievement of physics in the twentieth century has been to prove that this aim is unattainable.” – Jacob Bronowski. Can scientists ever be truly certain of their discoveries?
• Students studying more than one group 4 subject will be able to use these skills across all subjects
Aim 4: it is important that students see scientific errors and uncertainties not only as the range of possible answers but as an integral part of the scientific process
Aim 9: the process of using uncertainties in classical physics can be compared to the view of uncertainties in modern (and particularly quantum) physics

1.3 – Vectors and scalars

Essential idea: Some quantities have direction and magnitude, others have magnitude only, and this understanding is the key to correct manipulation of quantities. This sub-topic will have broad applications across multiple fields within physics and other sciences.
Nature of science: Models: First mentioned explicitly in a scientific paper in 1846, scalars and vectors reflected the work of scientists and mathematicians across the globe for over 300 years on representing measurements in three-dimensional space.
• Vector and scalar quantities Vectors and scalars Vectors and the NFL Vectors and Despicable Me
• Combination and resolution of vectors Vector addition Roger, Roger The Pythagorean theorem and the NFL Pronunciation of Pythagorean
Applications and skills:
• Solving vector problems graphically and algebraically
• Resolution of vectors will be limited to two perpendicular directions
• Problems will be limited to addition and subtraction of vectors and the multiplication and division of vectors by scalars
Data Booklet reference:
AH = A cos q
AV = A sin q
AH represents the horizontal component of the vector A. AV represents the vertical component of the vector A. q represents the angle between the vector A and the horizontal axis. A represents the magnitude of the vector A.
• Vector notation forms the basis of mapping across the globe
Theory of knowledge:
• What is the nature of certainty and proof in mathematics?
• Navigation and surveying (see Geography SL/HL syllabus: Geographic skills)
• Force and field strength (see Physics sub-topics 2.2, 5.1, 6.1 and 10.1)
• Vectors (see Mathematics HL sub-topic 4.1; Mathematics SL sub-topic 4.1)
Aim 2 and 3: this is a fundamental aspect of scientific language that allows for spatial representation and manipulation of abstract concepts


This is the complete problem set for Topic 1 - the same one I hand out. If you lose yours, you can download this one to replace it.


These are the Formative Assessments (practice) that you will do in order to prepare yourself for the Summative Assessments (evidence of proficiency). You can expect to receive a mark of at least Proficient on the Summative Assessment if you understand everything on these Formative Assessments.


Project marks are meant to replace summative assessment marks. Projects are your last opportunity to demonstrate your proficiency in meeting the standards of the assessment criteria.


Problem-solving strategies for suggested ways to attack word problems in physics. And for those purists among you, the genuine official SI reference document - all 188 pages of it!

Laboratory number crunching covers some of the methods you might encounter in the college level laboratory course. Least squares fit helps to develop a justification for the standard deviation which is presented in Laboratory number crunching. At present, standard deviation is not required by the IBO physics curriculum.

Unit vectors is a concise and elegant way to work with vectors. Most college courses in physics will use these things. Just skip the parts that look familiar and focus on the i, j, k and x,y,z-hats. Sorry, there is nothing on tensors, but they are even more concise.

Hamilton's formal introduction of vectors to the world is in this landmark paper published in 1849.

Fermi questions are named in honor of Enrico Fermi and are questions whose answers are found on napkins using estimation techniques and orders of magnitude.