A model ship is towed by two cables A and B which are at \(60^{\circ}\) to each other. A exerts a pull of \(20 N\) while \(B\) exerts a pull of \(30 N\). What is the resultant pull on the ship and the angle it makes with the \(30 N\) cable.
A. \(59 N ; 23^{\circ}\) B. \(44 N\); \(23^{\circ}\) C. \(44 ; 32^{\circ}\) D. \(43 N ; 23^{\circ}\)
Correct Answer: B
Explanation
by parallelogram law of vector Using cosine rule $$ \begin{aligned} &R^{2}=20^{2}+30^{2}-2(2)(30) \cos 120^{\circ} \\ &R^{2}=400+900-1200 \times\left(-\frac{1}{2}\right) \\ &R^{2}=1300+600 \\ &R^{2}=1900 \\ &R=\pm \sqrt{1900}=\pm 43.59 N \end{aligned} $$ To find the angle \((\theta)\) the resultant makes with the \(30 N\) cable (see figure C) Using sine rule, $$ \begin{aligned} \frac{\sin \theta}{20} &=\frac{\sin 120}{43-59} \\ \theta &=\sin ^{-1}\left[\frac{\sin 120}{43.59} \times 20\right] \\ \theta &=\sin ^{-1}[0.3974] \\ \theta &=23.4^{\circ} \end{aligned} $$