Basic Engineering Mathematics, Fifth Edition

Transposing formulae 85

9. I=PRT (T)


10. XL= 2 πfL (L)


11. I=

E

R

(R)

12. y=

x

a

+ 3 (x)

13. F=

9

5

C+ 32 (C)

14. XC=

1

2 πfC

(f)

12.3 Further transposing of formulae

Here are some more transposition examples to help
us further understand how more difficult formulae are
transposed.

Problem 9. Transpose the formulav=u+

Ft

m

to

makeFthe subject

v=u+

Ft

m

relates final velocityv, initial velocityu,

forceF,massmand timet.

(F

m

is accelerationa.

)

Rearranging gives u+

Ft

m

=v

and

Ft

m

=v−u

Multiplyingeach side bymgives

m

(

Ft

m

)

=m(v−u)

Cancelling gives Ft=m(v−u)

Dividing both sides bytgives

Ft

t

=

m(v−u)

t

Cancelling givesF=

m(v−u)

t

or F=

m

t

(v−u)

This shows two ways of expressing the answer. There
is often more than one way of expressing a trans-
posed answer. In this case, these equations forFare
equivalent; neither one is more correct than the other.

Problem 10. The final lengthL 2 of a piece of

wire heated throughθ◦C is given by the formula

L 2 =L 1 ( 1 +αθ)whereL 1 is the original length.

Make the coefficient of expansionαthe subject

Rearranging gives L 1 ( 1 +αθ)=L 2

Removing the bracket gives L 1 +L 1 αθ=L 2

Rearranging gives L 1 αθ=L 2 −L 1

Dividing both sides byL 1 θgives

L 1 αθ

L 1 θ

=

L 2 −L 1

L 1 θ

Cancelling gives α=

L 2 −L 1

L 1 θ

An alternative method of transposingL 2 =L 1 ( 1 +αθ)

forαis:

Dividing both sides byL 1 gives

L 2

L 1

= 1 +αθ

Subtracting 1 from both sides gives

L 2

L 1

− 1 =αθ

or αθ=

L 2

L 1

− 1

Dividing both sides byθgives α=

L 2

L 1

− 1

θ

The two answersα=

L 2 −L 1

L 1 θ

andα=

L 2

L 1

− 1

θ

look

quite different. They are, however, equivalent. The first

answer looks tidier but is no more correct than the

second answer.

Problem 11. A formula for the distancesmoved

by a body is given bys=

1

2

(v+u)t. Rearrange the

formula to makeuthe subject

Rearranging gives

1

2

(v+u)t=s

Multiplying both sides by 2 gives (v+u)t= 2 s

Dividing both sides bytgives

(v+u)t

t

=

2 s

t

Cancelling gives v+u=

2 s

t

Rearranging gives u=

2 s

t

−v or u=

2 s−vt

t

Problem 12. A formula for kinetic energy is

k=

1

2

mv^2. Transpose the formula to makevthe

subject

Rearranging gives

1

2

mv^2 =k