Forces (F_{1} (18N, 330°), F_{2} (10N, 090°)) and (F_{3} (25N, 180°)) act on a body at rest. Find, correct to one decimal place, the magnitude and direction of the resultant force.
Explanation
(F_{1} = (18N, 330°) ; F_{2} = (10N, 090°) ; F_{3} = (25N, 180°))
(R = F_{1} + F_{2} + F_{3})
= (begin{pmatrix} 18 cos 330 \ 18 sin 330 end{pmatrix} + begin{pmatrix} 10 cos 90 \ 10 sin 90 end{pmatrix} + begin{pmatrix} 25 cos 180 \ 25 sin 180 end{pmatrix})
= (begin{pmatrix} 18 cos 30 \ -18 sin 30 end{pmatrix} + begin{pmatrix} 10 times 0 \ 10 times 1 end{pmatrix} + begin{pmatrix} -25 \ 0 end{pmatrix})
= (begin{pmatrix} 18 times 0.866 \ -18 times 0.5 end{pmatrix} + begin{pmatrix} 0 \ 10 end{pmatrix} + begin{pmatrix} -25 \ 0 end{pmatrix})
= (begin{pmatrix} 15.588 – 25 \ -9 + 10 end{pmatrix})
= (begin{pmatrix} -9.41 \ 1 end{pmatrix})
(R = sqrt{(-9.41)^{2} + 1^{2}})
= (sqrt{88.5481 + 1})
= (sqrt{89.5481})
(R = 9.463N)
Direction: Let (theta) be the angle resultant makes with the x – axis.
(tan theta = frac{1}{-9.41} = -0.1062)
(theta = tan^{-1} (-0.1063) = -6.07°)
The resultant has magnitude 9.463N and makes an angle of 6.07° with the negative x- axis.