Conditions for equilibrium for two or three coplanar forces acting at a…

The idea

A point is in equilibrium when the resultant force on it is zero. For two coplanar forces, this requires equal magnitudes acting in exactly opposite directions along the same line. For three coplanar forces acting at a point, equilibrium requires the vector sum to be zero, which can be tested by resolving in two perpendicular directions and applying ΣFₓ = 0 and ΣF_y = 0, or by drawing the three forces tip-to-tail to form a closed triangle.

For example: A lamp of weight 30 N hangs from two strings each at 40° to the horizontal: resolving vertically gives 2T sin 40° = 30 N, so T ≈ 23.3 N. The three-force condition also means the lines of action of the forces must be concurrent if the body is to have no resultant moment.

What earns the marks

• State that equilibrium at a point means the resultant force is zero, so the vector sum of the forces is zero.
• For two forces, state that they must be equal in magnitude and opposite in direction along the same straight line.
• For three coplanar forces, resolve in two perpendicular directions and set the sum of components in each direction to zero.
• Draw the three forces tip-to-tail to form a closed triangle, showing that the resultant is zero.
• Use the closed triangle with trigonometry, including the sine rule or cosine rule, to find an unknown force or angle.
• Recognise that for three non-parallel coplanar forces in equilibrium their lines of action must meet at a common point.

Common mistakes and how to put them right

• Assuming three forces are in equilibrium whenever they form any triangle.

  Put it right: The corrected condition is a closed triangle with the force arrows arranged tip-to-tail so the resultant is zero.

• Treating the weight of a suspended object as acting along the string rather than vertically downwards.

  Put it right: The corrected free-body diagram shows weight acting vertically downwards from the point.

• Balancing forces in only one direction and ignoring the perpendicular direction.

  Put it right: The corrected method applies both ΣFₓ = 0 and ΣF_y = 0, or closes the vector triangle fully.

Exam tips

• Draw a free-body diagram of the point, showing each force with its correct direction and labelling unknown magnitudes clearly.
• Choose axes that simplify the algebra, for example horizontal and vertical, and mark each angle from the chosen axis before resolving.
• Check the final answer by substituting back into both component equations or by confirming the vector triangle closes.