3.1.3 Equilibrium in 2D

What does two dimensions (2D) mean in mechanics?

• 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 0xy 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

What is the resultant force in 2D and an unbalanced force in 2D?

• 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

What does equilibrium in 2D mean?

• 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

Worked Example

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

(b)  Find the value of F.

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