Equilibrium in 2D (AQA A Level Maths: Mechanics)

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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

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

Worked example

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.

3-1-3-fig2-we-solution-aqa-1

3-1-3-fig2-we-solution-aqa-a

(b)  Find the value of F.

3-1-3-fig2-we-solution-aqa-b

Examiner Tip

  •  Fnet 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.
  • Unless told otherwise, use g = 9.8 m s-2 rounding your final answer to 2 s. f.; if directed to use g = 10 m s-2  then round your final answer to one 1 s. f.

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Dan

Author: Dan

Expertise: Maths

Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.