Back to course
7.1MechanicsStretch

Static Particles

A particle in static equilibrium has no resultant force. By resolving forces in two perpendicular directions and setting each sum to zero, you get equations to find unknown forces.

24 min Applications of Forces
Edexcel A Level Maths: 7.1-7.2 Statics and ModellingCurated by Kepler, not produced by us. Watch it before the notes below.
Open on YouTube

What you'll be able to do

  • Apply equilibrium to a particle
  • Resolve forces in two directions
  • Set each resultant to zero
  • Solve for unknown forces
1

Equilibrium of a particle

A static particle has zero resultant force. Resolve all forces horizontally and vertically (or in any two perpendicular directions) and set each component sum to zero.

∑Fx=0,∑Fy=0\sum F_x = 0, \quad \sum F_y = 0
Equilibrium conditions for a particle.
Example 1A particle hangs in equilibrium on two strings. How do you find the tensions?
1Resolve horizontally: ΣFₓ = 0.
2Resolve vertically: ΣF_y = 0.
3Solve the two simultaneous equations.
Answer
Two equations from resolving in two directions.

Tip — For a particle (not a rigid body) you only need ΣF = 0 — moments are not required.

Formula recap

∑Fx=0\sum F_x = 0
Horizontal equilibrium.
∑Fy=0\sum F_y = 0
Vertical equilibrium.

Common mistakes to avoid

Taking moments for a particle.
A particle only needs ΣF = 0; moments are for rigid bodies.
Resolving in only one direction.
Resolve in two perpendicular directions for two equations.

Key takeaways

  • Static particle: zero resultant force.
  • Resolve in two perpendicular directions, each summing to zero.
  • No moments needed for a particle.

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

  1. A static particle has resultant force:

    • zero
    • maximum
    • equal to its weight
    • upward
    Show answer and explanation

    Answer: zero

    Equilibrium.

  2. For a particle in equilibrium you need:

    • ΣFₓ = 0 and ΣF_y = 0
    • ΣM = 0
    • only ΣFₓ = 0
    • a moment equation
    Show answer and explanation

    Answer: ΣFₓ = 0 and ΣF_y = 0

    Two force equations.

  3. Do you take moments for a particle?

    • No
    • Yes
    • Only on a slope
    • Only with friction
    Show answer and explanation

    Answer: No

    Particles have no size.

  4. Two unknown tensions need:

    • two equations
    • one equation
    • three equations
    • none
    Show answer and explanation

    Answer: two equations

    Resolve in two directions.

  5. Equilibrium equations come from:

    • resolving forces
    • differentiating
    • integrating
    • taking moments
    Show answer and explanation

    Answer: resolving forces

    ΣF = 0.

Next step

Keep practising

Start with the quiz above to check the key ideas. You can also choose another topic from the practice library.

Browse practice topics

Related lessons

All 6 lessons

Put the methods into practice: see how A-Level maths duels work. An account is required to play; the overview and sample are public.