H201 Measurements - 1. Notes (1718) [For Upload]

Measurement CONTENT 1. Physical Quantities and SI Units 2. Errors and Uncertainties 3. Scalars and vectors LEARNING OUTCOMES Candidates should be able to: a. recall the following base quantities and their units: mass (kg), length (m), (s), current (A), temperature (K), amount of substance (mol). b. express derived units as products or quotients of the base units and use the named listed in ‘Summary of Key Quantities, Symbols and Units’ as appropriate. c. Use of SI base units to check homogeneity of physical equations. d. show an understanding and use the conventions for labeling graph axes and table columns as set out in the ASE publication Signs, Symbols and Systematics (The ASE companion to 16-19 Science, 2000). e. 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). f. make reasonable estimates of physical quantities included within the syllabus. g. distinguish between scalar and vector quantities, and give examples of each. h. add and subtract coplanar vectors. i. represent a vector as two perpendicular components. j. Show an understanding of the distinction between systemic errors (including zero error) and random errors. k. show an understanding of the distinction between precision and accuracy. l. assess the uncertainty in a derived quantity by simple addition of actual, fractional or percentage uncertainties or by numerical substitution (a rigorous statistical treatment is not required). Physics Books for further reading : Physics for Scientists and Engineers. Serway. College Physics. Sears and Zemansky. Physics. Robert Hutchings. Physics. Tom Duncan.

1.0 Quantities and Units The science of physics is based upon taking measurements. All scientific theories and laws must be tested experimentally, and all experiments necessitate making measurements. 1.1 Physical Quantities: Base Quantities and Derived Quantities In Physics, quantities that can be measured are known as physical quantities. Each quantity consists of a numerical magnitude and a unit. (For vectors, there are directions as well.) Base quantities are physical quantities that are the most fundamental and they are independent of each other. The corresponding units for the base quantities are called the base units. It is important at all times to think of and write the value and the unit of any quantity together. The metric system of units was introduced at the time of the French Revolution to rationalise the chaos of units that existed at that time. It has been modified since then and now most countries use the metric system of units called the Systeme International (SI). The advantage of the SI system of units is that any quantity has only one unit in which it can be measured.  Quantities can be classified as base quantities or derived quantities.  Base quantities are the 7 physical quantities of the S.I. system by which all other physical quantities are defined. They arbitrarily chosen by scientists so that they: a) form the smallest set of physical quantities that will lead to a complete description of physics in the simplest terms. b) are based on international agreement by scientists.  Derived quantities are obtained from one or more of the base quantities through a defining equation. Base Units  There are 7 base units, one for each of the base quantities.  Base units are the 7 base units of the S.I. system , related to a base quantity, whose magnitude is defined without referring to any other units. The units used in measurement are the International System of Units (SI). Base Quantity Base Unit Name Length m metre Mass kg kilogram Time s second Electric current A ampere Temperature K kelvin Amount of substance mol mole Luminous Intensity cd candela

Derived Units  A derived unit can be expressed in terms of base units by using the defining equation of the quantity.  It is obtained from the base units by multiplication and/or division; wit hout including any numerical factors. Solution: The defining equation is E=hf f Eh Base units of 12 1 22     skgms skgmh Solution: 22 D FC vA Units of CD = )())(( )( 2213 2 mmskgm kgms   12 2    kgms kgms CD is dimensionless. (Quantities that have no units are known as dimensionless.) DERIVED QUANTITIES Defining Equation Base SI Units Derived Unit Volume V = l 3 m3 - Velocity v = d/t ms-1 - Frequency f = 1/T s-1 Hz Force F = ma Kgms-2 N Work or Energy W = Fd kgm2s-2 J Pressure P = F/A kgm-1s-2 Pa Charge Q = It As C Example 1: The energy of a photon of light frequency f is given by hf, where h is the Planck constant. What are the base units of h? Example 2: The drag coefficient CD of a car moving with speed v through air of density  is given by Apv FCD 22  where F is the drag force exerted on the car and A is the maximum cross - sectional area of the car perpendicular to the direction of travel. Show that CD is dimensionless.

Prefixes In physics, it is common to encounter quantiti es that are very large or minute in magnitude. For example, Earth’s mean radius is estimated to be 6 400 000 m. Radius of a hydrogen nucleus is approximately 0.000 000 000 000 001 3 m. To include all the zeros in all computation or steps will be undesirable and hence scientists adopted any of the 2 methods: use of scientific notation (standard form) or prefixes. Prefix Symbol Sub-multiple Prefix Symbol Sub-multiple Pico p 10-12 Kilo k 103 Nano n 10-9 Mega M 106 Micro μ 10-6 Giga G 109 Milli m 10-3 Tera T 1012 Centi c 10-2 Deci d 10-1 Solution 800, 000 W = 800 kW = 0.8 MW 0.0000325 m = 0.0325 mm = 32.5 m 2.65 x 10-10 s = 2.65 x 10-1 x (10-9) = 2.65 x 10-1 ns Example 3: Convert the following to suitable units with prefix: 800, 000 W; 0.0000325 m; 2.65 x 10-10 s

Estimates Of Physical Quantities You are expected to make reasonable estimates of the order of magnitude of common physical quantities. Solution The mass of an average person is about 60 kg and it consists of mainly water. Molar mass of water is 18 g = 0.018 kg. 60 kg consist of 60/0.018 = 3.3 x 103 moles. Hence no. of molecules = 3.3 x 103 x 6.02 x 1023 = 2 x 1027 molecules. Orders of Magnitude of Some Common Data Distances Radius of Earth 6400 km Moon-Earth Distance 3.8 x 108 m Earth-Sun Distance 1.5 x 1011 m Size of atom 10-10 m (1 Å) Size of nucleus 10-15 m Wavelength of visible light 4-7 x 10-7 m Density Density of air about 1 kgm-3 Density of water 1000 kgm-3 Density of metals about 103 kgm-3 Mass Mass of atom 10-27 kg Mass of person 60 kg Mass of earth 6.0 x 1024 kg Speed Speed of walking person ~ 3 kmh-1 or 0.8 ms-1 Speed of running person ~ 10 kmh-1 or 2.7 ms-1 Speed of car ~ 60 kmh-1 or 17 ms-1 Speed of molecular movement in air (Room temp) 300 ms-1 to 400 ms-1 Speed of light 3 x 108 ms-1 Speed of sound ~ 300 ms-1 Others Atmospheric pressure 1.01 x 105 Pa Room temperature 303 K (30 oC) Example 5: Which of the following gives the estimated number of atoms in your body? A 1024 B 1027 C 1030 D 1033 VJC 2010 Prelim Example 4: What is a reasonable estimate for the volume of a wooden metre rule found in a school laboratory? A 1.5 cm3 B 15 cm3 C 150 cm3 D 1500 cm3 Volume = 100 cm x 3 cm x 0.5 cm = 150 cm3 Answer C 2007 P1Q1

1.2 Homogeneity Of Physical Equations When we try to form equations that describe a physical system or phenomenon, it is important to understand how the co