Measurement Notes

 2013 Yeow Kok Han Page 1 Measurement 1 Importance of Measurement Scientific knowledge is powerful because it is not just untested theory and hypotheses. Scientists demand that the theories be supported with empirical evidence or measurements. For example, Einstein’s General Rela tivity theory suggested that gravity can bend the path of light but our ultimate confidence in the theory is whether the bending can be measured and checked against the amount predicted by the theory. Measurement is about quantifying things and the abilit y to quantify things allows calculations, analyses and deductions which in turn lead to new knowledge or theories. Once a theory is well supported by empirical data, it then can be used for predicting outcomes or results. To understand what the big deal about prediction is, let’s just consider the construction of a high rise building which costs millions of dollars. An architect will carry out calculations to make sure the designed building can withstand the expected loading and maybe possible earthquake. T hose calculations are done using theoretical formulae that have been verified with prior measurements! 2 SI System of Quantities & Units For measurements to be useful, they need to be expressed in terms of appropriate units which are internationally accepted. * See Joint Committee for Guides in Metrology (JCGM), International Vocabulary of Metrology, Basic and General Concepts and Associated Terms (VIM), III ed., Pavillon de Breteuil : JCGM 200:2012. International System of Quantities* consists of International System of Units* consists of 7 base quantities which cannot be defined in terms of other quantities: length, mass, time, electric current, thermodynamic temperature, amount of substance, luminous intensity 7 base units corresponding to the base quantities: metre (m), kilogram (kg), second (s), ampere (A), kelvin (K), mole (mol) and candela (cd) (Note: candela is not required in syllabus) Derived quantities are defined in terms of the base quantities. e.g. Force maF  but t va   and t sv   so F is finally defined in terms of the base quantities length, mass and time. Derived units are defined in terms of the base units. The notation* [Q] means ‘unit of Q’: [F] = [ma] = [m] [a] = [m] [v/t] = [m] [s/t]/[t] = kg m s-2 The derived unit kg m s -2 is given a more convenient short form N for newton. In the SI system, the same set of equations is used to express derived quantities or units in terms of the base quantities or units. Note also that quantities are always defined in terms of quantities and units are always defined in terms of units i.e. quantities and units are different entities. Hence it would be wrong to define speed(a quantity) as ‘the distance(a quantity) travelled per second(a unit)’. There are 7 base quantities and corresponding base units in the SI system of quantities and units. All other quantities are called derived quantities as they are ultimately defined in terms of the 7 base quantities using defining equations. The same equations are also used to work out the corresponding derived units. Quantities cannot be defined in terms of units and vice versa.

 2013 Yeow Kok Han Page 2 Examples of SI Derived Units Quantities Derived in terms of base and short form Energy kg m2 s-2 J Power m2 kg s-3 W Electric potential difference m2 kg s-3 A-1 V Electric resistance m2 kg s-3 A-2  Specific heat capacity m2 kg s-2 K-1 J kg-1 K-1 Electric field strength m kg s-3 A-1 V m-1 or N C-1 Multiples and submultiples. Frequently, there are situations where it is more convenient to use smaller or bigger units than the standard base and derived units. For that purpose, prefixes for multiples and submultiples are used together with the base or derived units. For example, when molecular size objects are being measured, it is more convenient to use nano - metre or nm in recordings. Another example is the use of mega-watt or MW when referring to power output of power stations. Examples of Non-SI units & Conversion Factors Length - inch () 1  = 0.0254 m or 2.54 cm Speed - knot (kn) 1 kn = 0.514444 m s-1 Energy - electronvolt (eV) 1 eV = 1.6  10-19 J Pressure - bar (bar) 1 bar = 105 Pa Time - minutes (min or ) 1 min = 60 s Time - hour (h or hr) 1 h = 3600 s Volume - litre (L or l) 1 L = 10-3 m3 or 1000 cm3 Without one, a measurement value is a pure number that does not tell us how much of what quantity we have and thus is a useless number. In addition, failure to pay attention to the kind of units used is a very common reason for students getting wrong values in calculations. Some people paid dearly for such mistakes. In 1999, NASA’s Mars orbiter smashed into the planet’s atmosphere because their engineers failed to convert English pound of force into SI newton in their calculations. Rule 1 – All terms in an equation must have the same units. Terms refer to quantities forming a group by multiplication or division and each group is separated from others by + - or = sign. For example, s = ut + ½ at2 is made up of three terms s, ut and ½ at2 and if s is to be in cm, then numerical values substituted must yield cm for both terms ut and ½ at2. SI derived units are often given short forms which one must learn to recognise. Prefixes for multiples and submultiples are often used and need to be learnt. The factors for multiples, submultiples and for converting non-SI to SI units are important in calculations. All terms in an equation must have the same units. Factor Prefix Name Symbol 10-12 pico p 10-9 nano n 10-6 micro  10-3 milli m 10-2 centi c 10-1 deci d 103 kilo k 106 mega M 109 giga G 1012 tera T Only those in syllabus

 2013 Yeow Kok Han Page 3 Rule 2 – Attention must be paid to the conversion factors when multiples or submultiples of base units or when non-SI units are used in calculations so that all terms will have consistent units. In contrast, using only SI base and SI derived units does not introduce any numerical factor into the equations. Examples: The net force required to give a 2 kg mass an acceleration of 3 m s-2 is calculated by Fnet = ma = 2(3) = 6 N but the net force required to give a 2 kg mass a n accel eration of 3 cm s -2 is definitely not calculated by Fnet = ma = 2(3) = 6 N. The latter case’s numerical value of 6 is not 6 N of force but should be 6 kg cm s-2 of force and 1 N is not the same as 1 kg cm s-2. Given a particle of mass 1.7  10-27 kg with KE 3 eV and desiring to find the speed in m s-1 using KE = ½ mv2, one must convert eV into J using the appropriate conversion factor. 3 Errors and Uncertainties In the majority of cases, the true value is unknown. However, many reference values have been established through careful measurements so that engineers and scientists can compare their measured values against these reference values instead. Measurement error is made up of two components: Systematic and Random. For calculations using numerical values in multiples or submultiples of units and non-SI units, their conversion factors should be used to make the units on both sides of the equation the same. Measurement error = Measured value – True value Measurement error = Systematic error (one which is constant or varies in a predictable way when the measurement is repeated) + Random error (one which varies in unpredictable manner when the measurement is repeated) Measured value Mean measured value Normal Distribution of measured values True value Systematic error Random