Answer to Question #240042 in Molecular Physics | Thermodynamics for Sakura

Question

Four sled dogs are pulling a 1000kg load. Sled dog A is pulling the load at 20N, 10 degrees north of east. Sled dog B is pulling the load at 55N, 70 degrees north of east. Sled dog C is pulling the load at 45N, 33 degrees north of west, while sled dog D exerts 30N at a direction of 80 degrees south of west. What is the resultant force acting on the load?

Answer 1

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

$F_{res,x}=20\ N\cdot cos10^{\circ}+55\ N\cdot cos70^{\circ}+45\ N\cdot cos(180^{\circ}-33^{\circ})+30\ N\cdot cos(180^{\circ}+80^{\circ})=-4.44\ N,$

$F_{res,y}=20\ N\cdot sin10^{\circ}+55\ N\cdot sin70^{\circ}+45\ N\cdot sin(180^{\circ}-33^{\circ})+30\ N\cdot sin(180^{\circ}+80^{\circ})=50.12\ N.$

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

$$F_{res}=\sqrt{F_{res,x}^2+F_{res,y}^2}=\sqrt{(-4.44\ N)^2+(50.12\ N)^2}=50.32\ N.$$

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

$$\theta=tan^{-1}(\dfrac{F_{res,y}}{F_{res,x}}),$$ $$\theta=tan^{-1}(\dfrac{50.12\ N}{-4.44\ N})=85^{\circ}\ N\ of\ W.$$