Basic Engineering Mathematics, Fifth Edition

148 Basic Engineering Mathematics

A graph ofyagainstx^2 is shown in Figure 18.1, with

the best straight line drawn through the points.Since a

straight line graph results, the law is verified.

53

y

A

C B

50

40

30

20

10

0 5 10 15 20 25

x^2

17

8

Figure 18.1

From the graph, gradient,a=

AB

BC

=

53 − 17

25 − 5

=

36

20

= 1. 8

and they-axis intercept,b= 8. 0

Hence, the law of the graph isy= 1. 8 x^2 + 8. 0

Problem 2. Values of loadLnewtons and

distancedmetres obtained experimentally are

shown in the following table.

Load,L(N) 32. 3 29.6 27.0 23.2

Distance,d(m) 0. 75 0. 37 0. 24 0. 17

Load,L(N) 18.3 12.8 10.0 6.4

Distance,d(m) 0. 12 0. 09 0. 08 0. 07

(a) Verify that load and distance are related by a

law of the formL=

a

d

+band determine

approximate values ofaandb.

(b) Hence, calculate the load when the distance is

0.20m and the distance when the load is 20N.

(a) ComparingL=

a

d

+bi.e.L=a

(

1

d

)

+bwith

Y=mX+cshows thatLis to be plotted ver-

tically against

1

d

horizontally. Another table of

values is drawn up as shown below.

L 32. 3 29. 6 27. 0 23. 2 18. 3 12. 8 10. 0 6. 4

d 0. 75 0. 37 0. 24 0. 17 0. 12 0. 09 0. 08 0. 07

1

d^1.^332.^704.^175.^888.^3311.^1112.^5014.^29

A graph ofLagainst

1

d

is shown in Figure 18.2.A

straight line can be drawn through the points,

which verifies that load and distance are related

by a law of the formL=

a

d

+b.

A

B C

L

35

31

25

20

30

15

10

11

5

0 2468101214

d

1

Figure 18.2

Gradient of straight line,a=

AB

BC

=

31 − 11

2 − 12

=

20

− 10

=− 2.

L-axis intercept,b= 35.

Hence, the law of the graph isL=−

2

d

+ 35.

(b) When the distanced= 0 .20m,

load,L=

− 2

0. 20

+ 35 =25.0N.

RearrangingL=−

2

d

+35 gives

2

d

= 35 −L and d=

2

35 −L