Forces in Two Dimensions

Forces are vectors and can be added and subtracted like any other vector.

Force vectors can be resolved into two perpendicular components in the x and y directions. 

The two force components are given by the following equations:

  • { F }_{ x }=Fcos\theta
  • { F }_{ y }=Fsin\theta

Example 1:

A force of 25N acts on an object at an angle of 30° above the horizontal. Calculate the horizontal and vertical components of this force:

Firstly, draw a diagram to represent the force and its components:

F_{ x }=25cos30^{ \circ }

F_{ x }=22.7N\quad right

F_{ y }=25sin30^{ \circ }

F_{ y }=12.5N\quad up

Example 2:

A force of 15N north and a force of 9N east act on an object. Calculate the resultant force acting on the object:

Firstly, draw a diagram to represent the forces:

To determine the resultant force:

{ F }_{ net }^{ 2 }={ F }_{ x }^{ 2 }+{ F }_{ y }^{ 2 }

{ F }_{ net }^{ 2 }={ 9 }^{ 2 }+{ 15 }^{ 2 }

{ F }_{ net }^{ 2 }=81+225

{ F }_{ net }^{ 2 }=306

{ F }=\sqrt { 306 }

{ F }=17.49N

Then we solve the angle, \theta

tan\theta =\cfrac { opp }{ adj }

tan\theta =\cfrac { 15 }{ 9 }

\theta =tan^{ -1 }\cfrac { 15 }{ 9 }

\theta =59.04^{ \circ }

Therefore, the resultant force is 17.49N at an angle of 59.04° in the east direction above the horizontal.