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1 Intoduction to Mathematics fo Enginees Fomulae Euations & Tansposition of EQUTION ND TRNPOITION OF FORMULE If at any stage you want to etun to whee you came fom use the Eploe Back Button. Intoduction Most of the Mathematics of Engineeing/cience consists of elationships between vaious physical uantities. These ae epessed as Mathematical euations. Eamples (i) ea of a cicle, adius (ii) Coulombs Law. The foce of attaction between two chaged paticles is F kqq whee Q, Q ae the chages, is thei distance apat, k is a physical constant. (iii) Rate of heat tansfe Q though a slab of heat conducting mateial, thickness is ( ) Q kt T 0 whee T T0 conductivity. the is tempeatue diffeence, k is the themal Euations such as the above which epesent feuently used esults ae known as fomulae. When using a fomula connecting physical uantitites it is of couse impotant to use a consistent set of physical units, but in these notes we ae solely inteested in the Mathematical ules fo manipulating euations.. Euations Thee is only one ule fo changing the appeaance of an euation. Whateve you do to one side of the euation you must do the same to the othe side
2 Intoduction to Mathematics fo Enginees Fomulae Euations & Tansposition of In the following L and R epesent whateve may appea on the left and ight sides of an euation espectively amd k epesents any numeical o algebaic uantity. If L R then (i) R L Eample If + y then + y The euation may be witten eithe way. Rathe than wite we would wite we would wite Rathe than wite (ii) L + k R + k Eample If y then y + + i.e. y + If you move a negative uantity fom one side to the othe it becomes positive. (iii) L k R k Eample If y+ then y+ i.e. y If you move a positive uantity fom one side to the othe it becomes negative. (iv) kl kr Eamples (a) If y +
3 Intoduction to Mathematics fo Enginees Fomulae Euations & Tansposition of then y ( + ) i.e. y ( + ) (b) If y + y + y+ This is sometimes called multiplying though by. (v) L k R k Eample If ( + y) then ( + y) + y This is called dividing though by. p p (vii) ( L) ( R) whee p is a ational numbe Eamples (i) If + y then ( ) + y + y (ii) If ( + y) then ( ) + y i.e. + y The following set of eamples ae concened with solving euations by making use of the above ules. Eamples
4 Intoduction to Mathematics fo Enginees Fomulae Euations & Tansposition of (i) (dd to each side) + 6 (ii) + (ubtact fom each side) (iii) + 6 (Multiply though by ) (iv) (v) (Divide though by ) (Invet both sides) ll the above ae eamples of LINER EQUTION. ny euation which can be witten in the fom a + b 0 whee a and b ae eal numbes is called a LINER EQUTION. The value of which satisfies the euation is called the oot of the euation. Linea euations b have one oot. a (vi) (uae both sides) (uae oot of both sides) ± (vii) (Multiply though by ) (Multiply though by ) ±
5 Intoduction to Mathematics fo Enginees Fomulae Euations & Tansposition of (i) + + (Multiply though by ) + 0 ( )( ) 0 o () It is vey tempting to divide though by BUT DON T 0 is a common facto ( ) 0 0 o If we had divided though by the oot at 0 would have been missed. Eamples (vi)-() ae eamples of QUDRTIC EQUTION. These ae coveed in detail in the notes FCTORITION ND QUDRTIC EQUTION. s we saw they may have two eal oots, one epeated oot, o no eal oots. (i) It is vey tempting to divide though by BUT DON T 0 is a common facto ( ) 0 The epession in backets factoises ( )( + ) 0 0 o o Eample (i) is a CUBIC EQUTION. The geneal fom is,, a b c d whee abcanddeal numbes. cubic euation may have thee eal oots o only one eal oot.thee is a fomula fo solving cubic euations but it is vey complicated and not much used instead we tend to use computational methods which will be coveed late in you couse. Tutoial In eecises -0 find the values of satisfying the euations ( + )( ) ( + ) + ( + ) 0 5
6 Intoduction to Mathematics fo Enginees Fomulae Euations & Tansposition of Click hee to go the solutions fo Tutoial. You can use the Eploe back button to etun hee.. Tansposition of Fomulae n impotant application of the above ules above ules occus in what is called tansposition of fomulae. We all know the fomula fo the aea of a cicle : Hence it is easy to calculate the aea of a cicle of adius 0.5m. (0.5) 0.75m But what if we ae told that the aea of a cicle is m and we need to know the adius? We need to ewite the fomula as to make the subject of the fomula. It is now uite easy to calculate the euied adius : m. This is called tansposing the fomula In the following eamples the aim is to make the given vaiable the subject of the fomula.. Rewite as F kqq F () kqq kqq F.Then and finally kq Q F. kt ( T0) ( T ) l Ql Ql k( T T ) then T T0 k and finally Ql T T0 + k. Q Fist 0. Fist a b + c then a b + c b ( ) b a c and finally b a c Let us look at this in moe detail. In the given fomula the following opeations ae caied out on the vaiable b :- (i) uae it (ii) dd c (iii) Take suae oot To makeb the subject we need to etace these steps. o we need to cay out the invese opeations in evese ode on the subject of the fomula as given which is a. Invese of (iii) gives a Invese of (ii) gives a c Invese of (i) gives This techniue is used in the net eample. a c 6
7 Intoduction to Mathematics fo Enginees Fomulae Euations & Tansposition of. R + ( R) Opeations on R :- (i) uae (ii) dd (iii) Divide by (iv) uae oot (v) Multiply by Invese opeations in evese ode (to be pefomed on ). (i) Divide by (ii) uae Gives (iii) Multiply by Gives (iv) ubtact Gives 6 Gives (v) uae oot Gives Theefoe the tansposed fomula is R It needs to be emphasised that the method shown above is of limited application as shown by the net eample.. RR R R + R ( R ) Befoe applying the above we would have to ewite the fomula so that the euied subject vaiable only appeaed once. In this case it is uicke and easie to poceed as follows: Fist RR+ RR RR then RR RR RR R( R R) finally RR R R R 5. gh This is a difficult poblem. We will do it two ways. Fist :- Divide both sides by. gh ( ) 7
8 Intoduction to Mathematics fo Enginees Fomulae Euations & Tansposition of uae both sides Invet both sides gh gh Multiply by gh gh gh + gh + Invet both sides gh + gh + nd finally gh + econd way :- Opeations on :- (i) Invet.(Raise to powe ), (ii), (iii) uae, (iv), (v) Invet, (vi) gh, (vii) uae oot (viii). (v) Invese in evese opeating on we obtain (i) gh + (vi) gh + (vii) gh + (ii) (viii) (iii) gh gh (iv) + gh Thus gh gh + + as befoe.
9 Intoduction to Mathematics fo Enginees Fomulae Euations & Tansposition of Tutoial. Given that a sphee has volume 0m find its adius.. Distance 5m 0 Time t secs P cceleation a ms - The diagam illustates a vehicle P which stats fom 0 with initial velocity constant acceleation ams. The vaious uantities ae elated by the fomula Given that ums, and ( ut s) a t a ms, u 0ms find s, the distance tavelled, afte 6 seconds.. In each of the following tanspose the given fomula to make the symbol in backets the subject of the fomula. (a) R RR R + R ( R ) (b) d h( h) ( ) (c) (d) V v (e) h( R + h ) 6 IR E IR gh ( R) ( I) ( ) Click hee to go the solutions fo Tutoial. You can use the Eploe back button to etun hee. Click hee to etun to the List 9
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