Equilibrium in 2D

In two dimensions, a particle may be considered as existing and being able to move around two-dimensional space rather than in a single straight line (1D)
- This 2D space is called a plane – so you may see the Oxy plane mentioned
The directions of the two dimensions are chosen so that they are perpendicular to each other
The two directions are usually
- horizontal and vertical, or
- parallel and perpendicular to an inclined plane

The resultant force in 2D is the sum of forces acting on a particle but this will now need to be considered in two parts – one for each direction
An unbalanced force is a force acting on a particle that whereby one or both parts are not cancelled by another force acting in the opposite direction
So a non-zero resultant force will be non-zero in at least one of the two dimensions, is therefore unbalanced and the particle will accelerate

In two dimensions a particle is in equilibrium if the resultant force acting on it is zero
- For example, if the two dimensions involved are the horizontal and vertical directions, a particle will be in equilibrium if any forces acting left are balanced by any forces acting right and any forces acting up are balanced by any forces acting down
When a particle has several forces (at least 3 – have a think as to why!) acting on it – and it is in equilibrium – the forces can be drawn “nose-to-tail” such that they form a polygon

3-1-3-fig1-four-forces-polygon

3.1.3_WE_Equilibrium in 2D_1

(a)

Write down the resultant forces acting on the particle in both the horizontal and vertical directions.

(b)

Find the value of

F

F_net is often used for resultant force in equations.
Sketching diagrams, or adding to any given in a question, can help you to understand a problem and pick up some marks.

Equilibrium in 2D (CIE AS Maths: Mechanics)