Additional Mathematics

Example 3

Forces of magnitudes 1 N, 2 N, 3 N, 4 Nand 5 N act along the lines OA, OB, OC, OD
and OE respectively, where OABCDE is a regular hexagon. Find the resultant of the
forces.. '

L

i

Fig. 23.14 A B

The forces acting are shown in Fig.23.14.
Take 1 and j along and perpendicular to OC respectively.

Sum of components along OC
= (I cos 60° + 2 cos 30° + 3 + 4 cos 30° + 5 cos 60°)1 = 11.21

Sum of components perpendicular to OC
= (-1 sin 60°-2 sin 30° + 0 + 4 sin 30° + 5 sin 60°)j = 4.5j

Hence the resultant forcer= 11.21 + 4.5j (Fig.23.15).

Fig. 23.15

4.5j r I

C2J

I

e

0 11.21

I r I'= 11.2^2 + 4.5^2 giving I r I= 12.1.

The angle 9 which the resultant makes with OC is given by

tan e = !1.^45 2 gtvmg · · e =^21.. 9°

Hence the resultant is 12.1 N-acting at 21.9° to OC.

We can also obtain the resultant graphically.

c

Let the forces along OA, OB, OC, OD, OE be a, b, c, d, e, respectively. A suitable
scale is chosen, say 1 em: 1 N. We s(arl by drawing a line to represent a in magnitude
and direction. (We can start with any force, the order is immaterial). Then we add the
other forces, one at a time, each force starting from the end point of the one before.