Question

A=2000 lb, north

B=4000 lb, 60 degrees north of west

C=5000 lb, 50 degrees west of south

D= 1000 lb, south

E= 3000 lb, 60 degrees of south

Find the resultant displacement using component method

Answer 1

Let's first find $x$- and $y$-components of resultant displacement:

$R_x=2000\ lb\cdot cos90^{\circ}+4000\ lb\cdot cos150^{\circ}+5000\ lb\cdot cos230^{\circ}+1000\ lb\cdot cos270^{\circ}+3000\ lb\cdot cos330^{\circ}=-4080\ lb,$

$R_y=2000\ lb\cdot sin90^{\circ}+4000\ lb\cdot sin150^{\circ}+5000\ lb\cdot sin230^{\circ}+1000\ lb\cdot sin270^{\circ}+3000\ lb\cdot sin330^{\circ}=-2330\ lb.$

We can find the magnitude of the resultant displacement from the Pythagorean theorem:

$$R=\sqrt{R_x^2+R_y^2}=\sqrt{(-4080\ lb)^2+(-2330\ lb)^2}=4698.5\ lb.$$

We can find the direction of the resultant displacement from the geometry:

$$\theta=sin^{-1}(\dfrac{R_y}{R}),$$ $$\theta=sin^{-1}(\dfrac{-2330\ lb}{4698.5\ lb})=30^{\circ}\ W\ of\ S.$$