Higher Engineering Mathematics, Sixth Edition

260 Higher Engineering Mathematics

158

458

2 m/s

4 m/s

3.5 m/s

308

Figure 24.32

10. An object is acted uponby twoforces of mag-
    nitude 10N and 8N at an angle of 60◦to each
    other. Determine the resultant force on the
    object.
    [15.62N at 26. 33 ◦to the 10N force]


11. A ship heads in a direction of E20◦Sata
    speed of 20knots while the current is 4knots
    in adirection ofN 30◦E.Determinethespeed
    and actual direction of the ship.
    [21.07 knots, E 9. 22 ◦S]

24.7 Vector subtraction

In Fig. 24.33, a force vectorFis represented byoa.

The vector(−oa)can be obtained by drawing a vector

fromoin the opposite sense tooabut having the same

magnitude, shown asobin Fig. 24.33, i.e.ob=(−oa)

b

2 F o

F a

Figure 24.33

For two vectors acting at a point, as shown in

Fig. 24.34(a), the resultant of vector addition is:

os=oa+ob.

Figure 24.33(b) shows vectors ob+(−oa),thatis,

ob−oaand the vector equation isob−oa=od.Com-

paringodin Fig. 24.34(b) with the broken lineabin

Fig. 24.34(a) shows that the second diagonal of the

‘parallelogram’ method of vector addition gives the

magnitude and direction of vector subtraction ofoa

fromob.

(a) (b)

a a

bsd b

o a o

Figure 24.34

Problem 11. Accelerations ofa 1 = 1 .5m/s^2 at

90 ◦anda 2 = 2 .6m/s^2 at 145◦act at a point. Find

a 1 +a 2 anda 1 −a 2 (i) by drawing a scale vector

diagram, and (ii) by calculation.

(i) The scale vector diagram is shown in Fig. 24.35.

By measurement,

a 1 +a 2 = 3 .7m/s^2 at 126◦

a 1 −a 2 = 2 .1m/s^2 at 0◦

a 1 a 2

a 1 a 2

2.6 m/s^2

Scale in m/s^2

1.5 m/s^2

145 

0 1 23

126 

a 1

a 2

a 2

a 1

Figure 24.35

(ii) Resolving horizontally and vertically gives:

Horizontal component ofa 1 +a 2 ,

H= 1 .5cos90◦+ 2 .6cos145◦=− 2. 13

Vertical component ofa 1 +a 2 ,

V= 1 .5sin90◦+ 2 .6sin145◦= 2. 99

From Fig. 24.36, magnitude ofa 1 +a 2 ,

R=

√

(− 2. 13 )^2 + 2. 992 =3.67m/s^2

In Fig. 24.36,α=tan−^1

(

2. 99

2. 13

)

= 54. 53 ◦and

θ= 180 ◦− 54. 53 ◦= 125. 47 ◦

Thus, a 1 +a 2 =3.67m/s^2 at 125. 47 ◦