CAIE A-Level · Mathematics 9709 · Forces and Equilibrium

Equilibrium of a Particle (9709 Mechanics 4.1)

11 min readSyllabus 4.1PreviewBy Uzair Khan

Syllabus objective

Use the principle that, when a particle is in equilibrium, the vector sum of the forces acting is zero, or equivalently, that the sum of the components in any direction is zero. Solutions by resolving are usually expected, but equivalent methods (e.g. triangle of forces, Lami's Theorem, where suitable) are also acceptable; these other methods are not required knowledge and will not be referred to in questions.

Introduction

When a particle is in equilibrium, it is either at rest or moving at constant velocity — crucially, its acceleration is zero. By Newton's second law (F=maF = ma), zero acceleration means the net force is zero. This is one of the most heavily examined ideas in 9709 Mechanics: virtually every statics problem — particles on slopes, objects hanging from strings, suspended signs — depends on it. Mastering equilibrium unlocks a systematic route to finding unknown forces, angles, and tensions under any configuration.

Core Concept

The Equilibrium Condition

A particle is in equilibrium if and only if the vector sum of all forces acting on it is zero.

Equivalently, for a 2-D system, the sum of components in every direction is zero. In practice this means choosing two convenient perpendicular directions (usually horizontal and vertical, or parallel and perpendicular to a slope) and writing two scalar equations:

∑Fx=0and∑Fy=0\sum F_x = 0 \qquad \text{and} \qquad \sum F_y = 0

These two equations allow you to solve for up to two unknowns (e.g. two tensions, or a force and an angle).

Strategy for Resolving

  1. Draw a clear force diagram labelling every force with its magnitude and the angle it makes with a chosen reference direction.
  2. Choose two perpendicular axes — align one axis with an unknown force where possible to eliminate it from one equation.
  3. Write the two resolving equations, being careful about signs (components in the positive axis direction are positive).
  4. Solve simultaneously if needed.

Other Acceptable Methods

The syllabus notes two alternative geometric methods. You are not required to know them, but they are valid if you choose to use them.

  • Triangle of forces: If exactly three forces act, they can be drawn tip-to-tail to form a closed triangle (since their vector sum is zero). Unknown magnitudes or angles are found using the sine rule or cosine rule on that triangle.
  • Lami's Theorem: For exactly three concurrent coplanar forces in equilibrium, each force divided by the sine of the angle between the other two is constant:
F1sin⁡α1=F2sin⁡α2=F3sin⁡α3\frac{F_1}{\sin\alpha_1} = \frac{F_2}{\sin\alpha_2} = \frac{F_3}{\sin\alpha_3}

where αi\alpha_i is the angle between the two forces other than FiF_i.

The standard examination approach remains resolving, and that is what is demonstrated throughout this note.

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Prerequisites: Components and Resultants of Forces

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