Access Mathematics

Transposition of Formulae

Learning objectives

 After this session you should be able to:  Recall simple formulae triangles to model simple engineering systems  Transpose formulae in which the subject is contained in more than one term  Transpose formulae which contain a root or a power 2

Recap: Make x the subject

Last lecture we examined the differences between equations and formulae and their subsequent solution protocols: Formula: Equation:

gx + h = k 3x+2 = 23

gx + h - h = k - h 3x+2 - 2= 23 -2 gx = k- h 3x = 23 - 2 x = (k - h)/g x = (23-2)/3 x=7 3

Transposition Of Formulae

 The rules are exactly the same as for algebra, except the final result is an algebraic expression instead of a numerical answer.

4

Simple Transposition

 In the Science units you will come across very simple formulae for instance  Newton’s second law (mechanics)   Electrical charge Ohms Law  Density

F

 

m V



ma V Q t

 

I IR

5

Recap: Simple Transposition

 Here the same rules apply as the letters in the formulae are just numbers in disguise

V R V

 

IR

;

I

&

R I V I



R I V



R

6

Activity

 In groups make the subject of the following formulae the variable in parenthesis for:  Density (

m

)  Electrical Charge (

t

)  Newton’s second law (

a

)  

m V Q



I t F



ma

7

Transposition of Elementary formulae  Mathematical

I P



PRT

100

P I PR



RT

100

R

100

I PR

 100

T

100

T

 100

I PR I



PRT

;

T

100  Systematic

I PR



T

100 100

I



T PR

Try this one yourselves: 2

r



c



c

 2 

r

;

r r



c

2  8

Extra terms

 Mathematical  v=u+at;t  v -u =u+at -u  v-u=at  (v-u) /a =at /a  t=(v-u)/a  Systematic  v=u+at;t  v -u =at  (v-u)/a=t Try this yourself but this time transpose for

a

instead 9

Transposition & substitution

Consider:

T



mv

2

r

;

r Tr



mv

2

r



mv

2

T

What if m=2, v=5 and T=10  Use any either of the methods to transpose find the the value of

R

given that: 

H



I Rt

.

10

Example:

 The pressure

p

of density  in a fluid at a depth

h

is given by:

p



p

 

g

h

Where

p a a

is the atmos pressure and

g

is gravitational acceleration.

 Make

h

the subject

p



p a

 

g

h p



p a

 

g

h



p

 

g

p a

 

h

11

Group Activities

  Work in groups Discuss the solution for

one

the following problems  Select a group member to share your solution with the class    

v



F

1

P



F

2 ;

F

1

R



R o

( 1  

t

);

t

1

P R

 

s



b

 100 ;

b

1

b



R

1 1

R

2 ;

R

1 12

Class Discussion/Exercise

 (a,b)

pV



nRT

;

P

,

T

   

R



r H

;

h h y



mx



c

;

m



I



R V



r

;

R

 

V

 

h

2 3  2

R



h

 ;

R R e

2  

R

1

R

1 

R

2

R

2

a

2 

b

2

a

2 ;

R

2 ;

a

2

A



a

(

H



h

) 

bh



cH

;

H

2 13

Subjects with Roots or Powers    In these cases we proceed as before isolating the power or the root first Thereafter we simply us the inverse operation in order isolate the required variable i.e. take the root or raise to the power respectively  e.g.

v

2 

u

2  2

as

;

u v

2  2

as



u

2 

v

2  2

as



u

Try 4

A

 

r

2

A

 

r

2 4

r

 4 

A

14

Transposition inc. Roots/Powers  The same procedure is employed where roots are involved. However to negate the root we raise to the appropriate power:  E.g.

T

 2 

L

;

L

g g



T

2  

L



g

2 

L

2

T

 or

T

2  2 

L

g

L



g

T

4  2 2 Try:

v



v

2  2

g

h

2

g

h

;

h v

2 2

g



h

15

Class Exercise

1 )

A

 

r

2 ;

r

2 )

E



mc

2 ;

c

3 )

E

 1 2

mv

2 ;

v

4 )

d

 5 )

E



f



A

;

A

2

V

; 2

U f

6 )

b



a

2 

c

2 ;

c

7 )

E

max 

m

g

h

 1 2

mv

2 ;

v

8 )

k



a

2 

b

2 ;

a

12 9 )

k

 10 )

k

 1 2

L

2 12 

R

2 ;

L

4

P

2 

Q

2 ;

P

16

Summary

 Have you met the learning objectives  Specifically are you

able

to:  Recall simple formulae triangles to model simple engineering systems  Transpose formulae in which the subject is contained in more than one term  Transpose formulae which contain a root or a power 17