Basic Engineering Mathematics, Fifth Edition

144 Basic Engineering Mathematics

(Note that the vertical lineABand the horizon-

tal lineBCmay be constructed anywhere along

the length of the straight line. However, calcula-

tions are made easier if the horizontal lineBCis

carefully chosen; in this case, 100.)

(b) TheR-axis intercept is atR=10ohms(by extra-

polation).

(c) Theequation of astraight lineisy=mx+c,when

yis plotted on the vertical axis andxon the hori-

zontal axis.mrepresents the gradient andcthe

y-axis intercept. In this case,Rcorresponds toy,

Vcorresponds tox,m= 1 .25 andc=10. Hence,

the equation of the graph isR=( 1 .25V+ 10 ).

From Figure 17.21,

(d) When the voltage is 60V, the resistance is 85 .

(e) Whentheresistanceis40ohms,thevoltageis24V.

(f) By extrapolation, when the voltage is 110V, the

resistance is 147 .

Problem 16. Experimental tests to determine the

breaking stressσof rolled copper at various

temperaturestgave the following results.

Stressσ(N/cm^2 ) 8.46 8.04 7.78

Temperaturet(◦C) 70 200 280

Stressσ(N/cm^2 ) 7.37 7.08 6.63

Temperaturet(◦C) 410 500 640

Show that the values obey the lawσ=at+b,

whereaandbare constants, and determine

approximate values foraandb. Use the law to

determine the stress at 250◦C and the temperature

when the stress is 7.54N/cm^2.

The co-ordinates (70, 8.46), (200, 8.04), and so on, are

plotted as shown in Figure 17.22. Since the graph is a

straight line then the values obey the lawσ=at+b,

and the gradient of the straight line is

a=

AB

BC

=

8. 36 − 6. 76

100 − 600

=

1. 60

− 500

=−0.0032

Vertical axis intercept,b= 8. 68

Hence, the law of the graph isσ= 0. 0032 t+ 8. 68

When the temperature is 250◦C, stressσis given by

σ=− 0. 0032 ( 250 )+ 8. 68 =7.88N/cm^2

Temperature t ( 8 C)

Stress

^

(N/cm

2 )

8.68

8.36

8.50

8.00

7.50

7.00

6.50

(^0) 100 200 300 400 500 600 700
6.76
y
x
B C
A
Figure 17.22
Rearrangingσ=− 0. 0032 t+ 8 .68 gives
0. 0032 t= 8. 68 −σ, i.e. t=
8. 68 −σ
0. 0032
Hence, when the stress,σ=7.54N/cm^2 ,
temperature,t=
8. 68 − 7. 54
0. 0032
=356.3◦C
Now try the following Practice Exercise
PracticeExercise 69 Practical problems
involving straight line graphs (answers on
page 347)

1. The resistanceRohms of a copper winding
   is measured at various temperaturest◦Cand
   the results are as follows:

R(ohms) 112 120 126 131 134

t◦C 20 36 48 58 64

Plot a graph ofR(vertically) againstt(hori-

zontally) and find from it (a) the temperature

when the resistance is 122and (b) the

resistance when the temperature is 52◦C.

2. The speed of a motor varies with armature
   voltage as shown by the following experi-
   mental results.