Forces and Equilibrium

2026 What You Need to Know (Syllabus Objectives)

By the end of this topic, you should be able to:

  1. Identify the forces acting in a given situation (e.g., by drawing a force diagram)
  2. Understand the vector nature of force, and find and use components and resultants
  3. Use the principle that when a particle is in equilibrium, the vector sum of the forces acting is zero
  4. Understand that a contact force between two surfaces has two components: the normal component and the frictional component
  5. Use the model of a 'smooth' contact and understand its limitations
  6. Understand limiting friction and limiting equilibrium, recall the definition of coefficient of friction, and use F = μR or F ≤ μR
  7. Use Newton's third law

1. What is a Force?

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A force is simply a push or a pull acting on an object. Forces can make things move, stop, speed up, slow down, or change direction.

The unit of force is the Newton (N), named after Sir Isaac Newton.

Drawing Force Diagrams

When solving mechanics problems, the first step is usually to identify all the forces acting on an object and draw a diagram showing them. This is called a force diagram or free body diagram.

Example: If you have a box sitting on a table with someone pushing it to the right with a force of 15 N, you would draw:

  • An arrow pointing right labeled 15 N (the push)
  • An arrow pointing down labeled W (the weight)
  • An arrow pointing up labeled R (the reaction from the table)

2. Weight

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Weight is the force of gravity pulling an object downwards towards the Earth.

Formula: W=mgW = mg

Where:

  • W = weight (in Newtons, N)
  • m = mass (in kilograms, kg)
  • g = acceleration due to gravity

Important: In mechanics problems, we always use g = 10 m/s² (not 9.81 as in physics).

Direction: Weight always acts vertically downwards (straight down, 90° towards the ground).

Example: A person has a mass of 113 kg. Find their weight.

Solution:

W=mg=113×10=1130 NW = mg = 113 \times 10 = 1130 \text{ N}

3. Forces as Vectors

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A vector is a quantity that has both magnitude (size) and direction. Force is a vector because it has:

  • A size (e.g., 10 N)
  • A direction (e.g., upwards, to the right, at 30° to the horizontal)

A scalar, by contrast, only has magnitude (size) but no direction. Examples include distance and speed.

Components of a Force

When a force acts at an angle, we can break it down into two perpendicular components (at right angles to each other). This is called resolving the force.

The Basic Rule: If a force F acts at an angle θ, then:

  • Horizontal component = F cos θ (the part acting left/right)
  • Vertical component = F sin θ (the part acting up/down)

Memory tip:

  • Adjacent to the angle → use cos
  • Opposite to the angle → use sin

Example: A force of 15 N acts at an angle of 40° to the vertical. Find the horizontal and vertical components.

Solution:

  • Vertical component = 15 cos 40° (because vertical is adjacent to the 40° angle from vertical)
  • Horizontal component = 15 sin 40° (because horizontal is opposite to the 40° angle from vertical)

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