RI Chap 1 Quantities and Measurement Lecture Notes

1 QUANTITIES & MEASUREMENT H2 Physics 9478 Content Page 1.1 Introduction 2 1.2 Quantities and Units 2 1.3 Prefixes, Standard Form and Significant Figures 6 1.4 Estimation 7 1.5 Measuring Instruments and Methods of Measurements 8 1.6 Systematic and Random Errors 8 1.7 Accuracy and Precision 10 1.8 Calculations of Uncertainties of Derived Quantities 11 1.9 Scalars and Vectors 16 1.10 Summary 21 Learning Outcomes Candidates should be able to: (a) recall and use the following SI base quantities and their units: mass (kg), length (m), time (s), current (A), temperature (K), amount of substance (mol) (b) recall and use the following prefixes and their symbols to indicate decimal sub- multiples or multiples of both base and derived units: pico (p), nano (n), micro (μ), milli (m), centi (c), deci (d), kilo (k), mega (M), giga (G), tera (T) (c) express derived units as products or quotients of the base units and use the named units listed in ‘Summary of Key Quantities, Symbols and Units’ as appropriate (d) use SI base units to check the homogeneity of physical equations (e) make reasonable estimates of physical quantities included within the syllabus. (f) show an understanding of the distinction between random and systematic errors (including zero error), which limit precision and accuracy (g) assess the uncertainty in derived quantities by adding absolute or relative (i.e. fractional or percentage) uncertainties or by numerical substitution (rigorous statistical treatment not required). (h) distinguish between scalar and vector quantities, and give examples of each (i) add and subtract coplanar vectors (j) represent a vector as two perpendicular components

Page| 2 1.1 Introduction Physics is an experimental science. Precise and accurate measurements enable the collection of useful experimental data that can be tested against theoretical predictions to refine the development of physical theories. Experimental evidence is the ultimate authority in discriminating between competing physical theories. Scientific knowledge continues to evolve as data from new or improved measurements helps us to better understand and explain physical phenomena. Measurements are subject to uncertainties , and it is important to estimate these to understand the reliability of the measurements. Error analysis involves estimating the uncertainties in measurements and finding ways to reduce them if necessary. In an experiment, the record of measurements made should include the estimated uncertainties and an analysis of the possible sources of errors with a discussion of steps taken to reduce the uncertainties should be documented . Doing this enables better conclusions to be drawn from the experimental data. The act of measurement affects the object being measured due to the interaction between the measuring device and the object. Common examples of this include measurements made using a thermometer, voltmeter or ammeter. Thus, improving the accuracy of measurements often requires the use of better instruments and enhanced experimental techniques. Physicists are very serious about measurements, and the other sciences and society as a whole have benefitted from the spill-over effects of the invention of many amazing measuring devices and techniques. Modern engineering also depends heavily on acc urate measurements in areas such as design, construction, optimiz ation and communication. Precise measurements have made many advanced technological applications possible; examples include the study and manipulation of materials, and breakthroughs in fields as diverse as geophysics and biology. Measurements using sophisticated devices like magnetic resonance imaging (MRI) scanners are important in the medical industry as it provides a wealth of data that aids in clinical diagnosis and influences decisions with regards to treatment. 1.2 Quantities and Units  Physical Quantities The laws of P hysics are commonly expressed as mathematical relationships among physical quantities and are verified by conducting experiments which involved the measurements of these quantities. The magnitude of a physical quantity is expressed as a value together with a unit of measurement. All physical quantities consist of a numerical value and a unit. For “a force of five newtons”, force ‘F’ is the physical quantity, ‘5 N’ is the magnitude with newtons ‘N’ being the unit.

RAFFLES INSTITUTION YEAR 56 PHYSICS DEPARTMENT 3 | Page  S.I. Base Quantities and Units Base quantities are physical quantities that are the most fundamental and they are used to define other physical quantities. Scientists, in the interest of simplicity, chose seven base quantities that give a full description of the physical world. The Système Internationale d’Unités (International System of Units or SI units) is based on the seven base quantities , and their corresponding base units are listed below. In 2019, some of SI base unit s were redefined. (Use the Internet to find out what they are.) Base quantity Base unit Symbol time second s length metre m mass kilogram kg current ampere A temperature kelvin K amount of substance mole mol luminous intensity (not in syllabus) candela cd International System of units The True Measure of a Kilogram The International Atomic Time  S.I. Derived Quantities and Units Derived quantities are physical quantities formed by combining base quantities accord ing to algebraic relations involving products and/or quotients. Hence, derived units are defined in terms of base units and are expressed as products and/or quotients of base units. Derived units are obtained from the base units according to a defining equation that relates the physical quantities. For example, the defining equation for speed v is given by = sv t , where s is distance and t is time. Hence, the unit of speed is metre per second, i.e. m s−1. NASA Mars Orbiter Unit Conversion Mistake There is a space between the m and the s −1. Otherwise, it may be misread as per millisecond (ms-1). Units should always be expressed in indices form, e.g. m s−2 instead of m/s2. Other examples of derived units are given below. Derived quantity Defining equation Base units Derived unit Symbol of derived unit force force = mass × acceleration (kg) × (m ÷ s ÷ s) = kg m s–2 newton N work done work done = force × displacement (kg m s–2) × (m) = kg m2 s–2 joule J magnitude F = 5 N unit physical quantity value

Page| 4  Homogeneity of Equations For a physical equation to be operational or meaningful, each term in the equation must have the same base units (or dimensions). Only quantities with the same base units can be added, subtracted or equated. In other words, each term separated by “+”, “-” or “=” sign must have the same base units. When each of the terms in a physical equation has the same base units, the equation is said to be homogeneous or dimensionally consistent. Example 1 Analyse whether the equation v u at= + is homogeneous. Example 2 Analyse whether the equation 22 2 2v u as= + is homogeneous. Checking the homogeneity of an equation using base units is a powerful way of establishing if the physical equation is plausible. See Example 4.

RAFFLES INSTITUTION YEAR 56 PHYSICS DEPARTMENT 5 | Page Example 3 Analyse whether the equation 2s ut at= + is homogeneous. However, we know that the correct equation that describes how the displacement of an object moving with constant acceleration varies with time is 21 2s ut at= + . Hence, the equation 2s ut at= + is homogenous but is physically wrong! Note: An equation which is found to be homogenous need not be physically correct due to either (i) wrong coefficients/signs, o