Answer to Question #241008 in Physics for Bonbon

Question

Scalar and Vector Quantities


Two forced A and B acting on a particle. Force A is 3 N directed north and B is 4 newton, 30 degrees of east. Find the resultants force using the laws of sines and cosine.

Answer 1

If the second force points 30° N of E, the angle between them is θ=90°-30°=60°. The angle between their lines, counting clockwise, is γ=180°-60°=120°.

According to the law of cosines, the resultant will be

$$R=\sqrt{A^2+B^2-2AB\cos\gamma}=\\ =\sqrt{3^2+4^2-2·3·4\cos120°}=6.1\text{ N}.$$

The angle toward E from N, according to law of sines, is

$$\frac{B}{\sin\beta}=\frac{R}{\sin\gamma},\\\space\\ \beta=\arcsin\bigg(\frac BR\sin\gamma\bigg)=34.6°.$$