Transposition of formulae - Mathematics resources

1 Transposition offormulaemc-TY- Transposition -2009-1In Mathematics , engineering and science, formulae are usedto relate physical quantities to eachother. They provide rules so that if we know the values of certain quantities, we can calculatethe values of others. In this unit we discuss how formulae canbe transposed, or transformed, order to master the techniques explained here it is vital that you undertake plenty of practiceexercises so that they become second reading this text, and/or viewing the video tutorial on this topic, you should be able to: transpose formulae in order to make other variables the subject of the a simple linear of simple formula for the simple examples of useful mathcentre 20091. IntroductionConsider the formula for the period,T, of a simple pendulum of lengthl:T= 2 lg,wherelis the length of the pendulumNow, on Earth, we tend to regardg, the acceleration due to gravity, as being fixed.

2 It varies alittle with altitude but for most purposes we can regard it asa suppose we had a pendulum of a fixed lengthl, and we took it somewhere else, say themoon, or Mars. The gravity there would not be the same and so the value ofgwould be differentfrom its value on we wanted to measureg. One way would be to take the pendulum, set it swinging andmeasure the time,T, for one complete cycle. Once we have measured the time, we could thenuse that to calculateg. But before we could do this, we would need to know whatgwas, interms of the rest of the symbols in the formula. So we need to re-arrange the formula so thatit states g=? . We will show you how to do this rearrangement later in the rearrange, transform, or tranpose the formula, we need many of the techniques used to solveequations. So, the video Solving Linear Equations in One Variable might be very useful to havea look Solving a simple linear equationBefore we look at rearranging more complicated formulae we recap by having a look at a simplelinear equation.

3 Suppose we wanted to solve3x+ 5 = 6 3(5 2x)Our aim is to end up with an expression forx, that is x=? . We start by expanding the bracketson the + 5 = 6 15 + 6x3x+ 5 = 6x 9We then proceed to manipulate this to try to get all the terms involvingxon to one side. Wemust preserve the balance in the original equation by doing exactly the same operations to both sides:3x 3x+ 5 = 6x 3x 95= 3x 9 Adding 9 to both sides:5 + 9 = 3x 9 + 914 = 3xand sox=143We have obtained an expression forxas mathcentre 2009 With practice the amount of working written down will reducebecause you will be able to carryout many of the stages at the same time. Having gone through the steps of solving an equation,the same technique, particularly the idea of keeping the balance bydoing the same thing to bothsides, is what we are going to do when we look at the transformation of Transposition of simple formulaeExampleConsider the formulav=u+at.

4 Suppose we wish to transpose this formula to obtain one we want to obtainton its own we start by subtractingufrom each side:v=u+atv u=atWe now divide everything on both sides ua=ata=tand so finallyt=v ua. We have transposed the formula to find an expression the formulav2=u2+ 2asand suppose we wish to transpose it to want to obtainuon its own and so we begin by subtracting2asfrom each + 2asv2 2as=u2 Finally, taking the square root of both sides:u= v2 2asNotice we need to take the square root of the whole term ( v2 2as) in order to the formulas=ut+12at2. Suppose we want to transpose it to we wantaon its own, we begin by subtractingutfrom both +12at2s ut=12at2 Multiplying both sides by 2:2(s ut) =at2 Dividing both sides byt2:2(s ut)t2=aand soa=2(s ut) mathcentre 2009 ExampleSuppose we wish to rearrangey(2x+ 1) =x+ 1in order to thatxoccurs both on the left and on the right.

5 We need to try to get all the termsinvolvingxtogether. We begin by expanding the brackets on the left:y(2x+ 1) =x+ 12xy+y=x+ 1 Subtractingxfrom both sides:2xy x+y= 1 The left-hand side now has two terms involvingx. We can factorise these as follows:x(2y 1) +y= 1 Then subtractingyfrom both sides:x(2y 1) = 1 yand finally, dividing both sides by(2y 1)x=1 y2y 1 ExampleSuppose we wish to rearrangeyy+x+ 5 =xto find an expression begin by multiplyingevery termon both sides by(y+x)in order to remove the fractions:y+ 5(y+x) =x(y+x)Next we multiply out the brackets:y+ 5y+ 5x=xy+x2We try to get all the terms involvingyonto the left-hand side. Subtractingxyfrom both sides:6y xy+ 5x=x2 Subtracting5xfrom both sides, and taking out the common factorywe havey(6 x) =x2 5xFinally, dividing both sides by6 xwe obtainy=x2 5x(6 x) mathcentre 2009 Exercise 1 Rearrange each of the following formulae to make the quantity shown the +at, + 2as, 12at2, 2(w+h), 2 r2+ 2 rh, +mgh, +mgh, (3b 1) = 2b+ 2, s= 3s, s+ 5 = 3t,s4.

6 The formula for the simple pendulumWe began with the formulaT= 2 lg. Let us now try to rearrange this to find an begin by squaring both sides of the equation in order to remove the square (2 )2lgTo remove the fraction we multiply both sides byg:T2g= (2 )2lDividing both sides byT2givesg=(2 )2lT2By observing the two square terms on the right, we note that this formula could be written, ifwe wish, in the equivalent formg=(2 T) mathcentre 20095. Further examples of useful formulaeExample - the lens formulaThe so-called lens formula, which is used in optics, is givenby1f=1u+1vSuppose we want to rearrange this formula to we want to isolateuwe begin by subtracting1vfrom both 1v=1uThe left-hand side fractions can be combined by expressing them over a common denominatorv ff v=1uInverting both sidesf vv f=uand sou=f vv fas formulaT=T0(1 v2c2)1/2arises in the study of relativity.

7 Suppose we want to rearrange itto find an expression begin by noticing that if we square both sides this will remove the square root term ( thepower12) on the right-hand side. So squaring:T2=T20(1 v2c2)We remove the fraction by multiplying both sides by(1 v2c2):T2(1 v2c2)=T20 Dividing both sides byT2:(1 v2c2)=T20T2 Addingv2c2to both sides gives1 =T20T2+ mathcentre 2009 SubtractingT20T2from both sides:1 T20T2=v2c2 Finally, taking the square root of both sidesvc= 1 T20T2as 2 Rearrange each of the following formulae to make the quantity shown the +1x, +11 x, r2, a(1 x), r2 1,rAnswersExercise (vt s) (p 2w) 2 r22 2(E mgh) + +a3a 10t3t 1 1y 1 1 1a(mk) 1 +(V0V) mathcentre 2009