NUMBERS, MATHEMATICS AND EQUATIONS

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1 AUSTRALIAN CURRICULUM PHYSICS GETTING STARTED WITH PHYSICS NUMBERS, MATHEMATICS AND EQUATIONS An integral part t the understanding f ur physical wrld is the use f mathematical mdels which can be used t explain physical phenmena and be used t make predictins. The heart f a mathematical mdel are its equatins. It is surprising that even a simple equatin such is really a very cmplex statement. The equal sign simply means the number f the left is equal in magnitude t the number n the right. On the left hand side represents a net frce but the prduct n the right des nt. BASIC MATHEMATICS Using yur calculatr Yu will need t be cmpetent in using yur calculatr fr a wide range f scientific calculatins invlving expnentials, lg functins and trig functins (degrees and radians), and be able t display numbers in scientific ntatin. Significant figures A measurement is the result sme prcess f bservatin r experiment. The aim f any measurement is t estimate the true value f sme physical quantity. Hwever, we can never knw the true value and s there is always sme uncertainty assciated with the measurement (except fr sme simple cunting prcesses). A rugh methd f indicating the degree f uncertainty is thrugh quting the crrect number f significant figures. The usual cnventin is t qute n mre than ne uncertain figure. S when yu write dwn a number, the last figure in that number shuld be the ne that is in dubt. Rules fr assigning significance t a digit Digits ther than zer are always significant. Final zers after a decimal pint are always significant. Zers between tw ther significant digits are always significant. Zers at the end f a number maybe ambiguus in cunting the number f significant figures. AUSTRALIAN CURRICULUM PHYSICS 1

2 Example 1 In a lab activity, fur students calculated the mass f a brass blck. The measurements they recrded were M 1 = g M 2 = 2002 g M 3 = 2000 g M 4 = g Measurement 1 is given t 11 significant figures, the recrding f such a measurement is ridiculus. The mass culd nt be calculated t this number f significant figures. Measurement 2 has 4 significant figures. Measurement 4 has 5 significant figures. But, what abut measurement 3 it is ambiguus. The best way t clearly indicate the crrect number f significant figures is t write the number in scientific ntatin with ne digit t the left f the decimal place, s fr measurement 3, we culd write M 3 = g (1 significant figure) M 3 = g (2 significant figures) M 3 = g (3 significant figures) M 3 = g (4 significant figures) A measurement such as 2000 g has an ambiguus number f significant figures, but withut any ther infrmatin, yu can assume that it has 4 significant figures in ding a calculatin. Example 2 Remember, the last digit is usually the ne in dubt. Fr example, yu prbably knw yur height t a few centimetres and yu culd write it as h = 1.73 m 3 is uncertain. Fr a tall friend f yurs, yu can nly guess their height and s yu wuld recrd h = 1.9 m 9 is the dubtful number. Example 3 (a) (3 significant figures since = ). (b) (5 significant figures). (c) (4 significant figures). Example 4 Cunting numbers 101, 102, 103, have an unlimited number f significant figures(sf). Example 5 additin r subtractin = AUSTRALIAN CURRICULUM PHYSICS 2

3 Example 6 multiplicatin r divisin In multiplicatin and divisin, the result shuld have n mre significant figures than the number having the fewest number f sf. Fr example, has nly 3 significant digits, and has 5. S accrding t the rule the prduct answer culd nly be expressed with 3 significant digits = Example 7 square rt The rt r pwer f a number shuld have as many significant figures as the number itself = Recrding a measurement f a physical quantity The best way t recrd a measurement is t use ne f the fllwing frmats: (1) name f physical quantity, symbl (ften used with subscript) = value ( uncertainty) unit (2) name f physical quantity, symbl (ften used with subscript) = value unit Fr the final recrding f yur measurement, the last digit in any number is the ne which in dubt. Small and large values shuld always be written in scientific ntatin. Example 1 The measurement f Ian s waist can be recrded as L = 0.87 m the digit is 7 is in dubt L = mm the digit is 7 is in dubt (870 mm is misleading) L = m the digit is 5 is in dubt L = mm the digit is 5 is in dubt L = ( ) m uncertainty is 0.01 m (10 mm) L = ( ) 10 2 mm uncertainty is 0.01 m (10 mm) L = ( ) m uncertainty is m ( 5 mm) L = ( ) 10 2 mm uncertainty is m ( 5 mm) L = mm incrrect can nt measure waist t < 1 mm Example 2 Ian s height, h 1 = 1.70 m Jan s height, h 2 = ( ) m Ian s mass, m 1 = 65.2 kg Jan s mass, m 2 = ( ) 10 3 g speed f light, c = m.s -1 charge n electrn, e = C AUSTRALIAN CURRICULUM PHYSICS 3

4 Quadratic equatin yu shuld be able t slve a quadratic equatin ax bx c 0 x b b ac 2a Trignmetry Fr a right angle triangle with sides a, b and c (hyptenuse) and with angles A, B and C = 90 sin A a cs A b tan A a c c b sin A cs A 2 2 tan A sin A cs A 1 a B C b c A Pythagrean therem Area f triangle = ½ a b c a b Gemetric frmulas Circumference f a circle (radius r), C = 2 r Area f a circle (radius r), A = r 2 Area f sphere (radius r), A = 4 r 2 Vlume f sphere (radius r), V = (4/3) r 3 Area f rectangle (sides L 1 and L 2 ), A = L 1 L 2 Vlume f blck (sides L 1, L 2 and L 3 ) V = L 1 L 2 L 3 Vlume f a right cylinder (height h and radius r) V = r 2 h Rearranging an equatin A very imprtance and essential skills is t be able t rearrange an equatin. This can be difficult but if yu fllw a well define prcedure yu will be able t master this skill. Always rewrite the equatin with the quantity yu want n the left hand side f the equals sign and then perfrm a series f mathematical peratins t bth sides f the equatin. If yu have any difficulties, then d the peratins step by step. Example 1 Find an expressin fr frm the equatin P A T T AUSTRALIAN CURRICULUM PHYSICS 4

5 Slutin Identify / Setup T =? n left hand side f equatin Rearrange equatin step by step Execute A T T P T T P A T P T A T T 4 P A 1 4 Evaluate T n left k T term k Changing units It is ften necessary t cnvert ne set f units int anther. This can be dne by reducing the cnversin t a simple algebraic prblem. The fllwing examples will illustrate hw t d this. Example 1 A car is travelling at a speed f 165 km.h -1. What is the speed f the car in m.s -1? Identify / Setup Basic cnversins 1 km = 10 3 m 1 h = (60)(60) s = s 1 km. h -1 = (10 3 ) / ( ) m.s -1 Execute 165 km.h -1 = (165) (10 3 ) / ( ) m.s -1 = 45.8 m.s -1 Evaluate answer shuld be smaller, k AUSTRALIAN CURRICULUM PHYSICS 5

6 Example 2 The density f a liquid was 1.8 g.ml -1. What is the density in kg.m -3? Slutin Identify / Setup Basic cnversins 1 g = 10-3 kg 1 ml = 1 cm 3 1 cm = 10-2 m 1 cm 3 = (10-2 ) 3 cm 3 = 10-6 m 3 1 g.ml -1 = (10-3 ) / (10-6 ) kg.m -3 Execute 1.8 g.ml -1 = (1.8) (10-3 ) / (10-6 ) kg.m -3 = kg.m -3 Evaluate Answer shuld be bigger e.g. density f water 1 g.ml -1 r 1000 kg.m -3 k EQUATION TEMPLATES One needs t develp a methd fr getting behind the cmplexity f equatins. One way f ding this is cnstructing equatin templates t gain a better understanding f equatins. An equatin template invlves cmpleting a detailed summary f the stry that the equatin tells alng the lines: State what the symbls represent (meaning & interpretatin), S.I. units, ther units, typical values, vectr r scalar, psitive r negative quantity. A visualisatin f what the equatin is abut, is it a definitin r a law, when is it applicable, cmments and an interpretatin. Alternative frms f the equatin. Graphical representatins f the equatin. Numerical examples. Examples f equatin templates will ften be given thrughut the web ntes. AUSTRALIAN CURRICULUM PHYSICS 6

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