Five coplanar forces, each of magnitude F, are acting on a particle. Each force is inclined at an angle `30^@` with the previous one. Find out the magnitude and direction of the resultant force on the particle.

Explanation

Let us choose x-axis along the first force , and y-axis along the perpendicular on the plane of the forces [Fig.2.34].
The angles made by the five forces with the x-axis are `0^@ ,30^@,60^@,90^@ and 120^@`.

`therefore` Component of resultant force along x-axis,
`F_x= F cos 0^@+F cos 30^@+F cos 60^@+ F cos 90^@ +F cos 120^@`
`=F[1+sqrt(3)/2+1/2+0-1/2]`
`=F(1+sqrt(3)/2)`
and component resultant force along y-axis ,
`F_y=F sin 0^@ +F sin 30^@+ F sin 60^@ +Fsin 90^@ +F sin 120^@`
`=F[0+1/2+sqrt(3)/2+1+sqrt(3)/2]=F(3/2+sqrt(3))`
`=sqrt(2) F (1+sqrt(3)/2)`
So, the magnitude of the resultant force,
`F.sqrt(F_(x)^(2)+F_(y)^(2) =F(1+sqrt(3)/2) sqrt(1^2+(sqrt(3))^2)`
`=2F(1+sqrt(3)/2)=(2+sqrt(3))F`
The angle of inclination of F. with the x-axis ,
`theta =tan ^(-1)(F_y)/(F_x) =tan ^(-1)sqrt(3) =60^@` (along the third force )