Problems may be solved either by the use of resolved forces or the use of…

The idea

When several coplanar forces act at a point, you can analyse equilibrium in two equivalent ways. Resolving splits every force into perpendicular components, usually horizontal and vertical, then applies ΣFₓ = 0 and ΣF_y = 0. The closed-triangle method draws the forces tip-to-tail; if they form a closed triangle, the resultant is zero. For example, a 50 N weight on a string pulled sideways by a horizontal force with the string at 30° to the vertical: resolving gives T cos 30° = 50 N and T sin 30° = F, so T ≈ 57.7 N and F ≈ 28.9 N. A closed triangle with the same three forces gives the same values by trigonometry. Choose whichever method suits the geometry, but show the full vector diagram or component equations so the reasoning is clear.

What earns the marks

• State that a body in equilibrium has zero resultant force, so the vector sum of all forces acting at the point is zero.
• Resolve each force into two perpendicular directions, commonly horizontal and vertical, using F cos θ and F sin θ with θ measured from the chosen axis.
• Apply ΣFₓ = 0 and ΣF_y = 0 simultaneously to form solvable equations for unknown magnitudes or angles.
• Draw a closed vector triangle with forces placed tip-to-tail, ensuring each arrow direction represents the force acting on the point.
• Use sine rule, cosine rule or right-angled trigonometry on the closed triangle to determine an unknown force or angle.
• Check that the answer is physically sensible, including units, direction and relative size compared with the given forces.

Common mistakes and how to put them right

• Using the wrong trigonometric ratio for a component.

  Put it right: The component along an axis is F cos θ when θ is measured from that axis, and F sin θ when θ is measured from the perpendicular axis. Always mark θ on the diagram before choosing.

• Drawing the closed triangle with arrows tail-to-tail rather than tip-to-tail.

  Put it right: The corrected diagram has each force arrow starting where the previous one ends, and the triangle closes only when the resultant is zero.

• Forgetting that the two resolution equations must be solved together.

  Put it right: Substituting a value from one equation into the other, or ignoring one direction, loses the link between the unknowns.

Exam tips

• Sketch the forces at the point and label every angle relative to a clearly chosen axis before writing any equation.
• If the geometry is a right-angled triangle, resolving is usually quicker; if three forces form a scalene triangle, the sine rule or cosine rule may be more efficient.
• Keep full calculator precision through intermediate steps and round only the final answer to an appropriate number of significant figures.