Represent equilibrium with a force triangle
Draw three coplanar forces head to tail and use a closed vector triangle to represent zero resultant force.
AS Level; Cambridge International Physics 9702, 2025–2027 syllabus, Topic 4. These original explanations and numerical practice support theory study; they do not demonstrate supervised practical competence. Full timed assessment and human subject-teacher review remain outstanding.
Before you begin
- Use SI units, resolve perpendicular components, and distinguish force from pressure.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What does a closed head-to-tail triangle of forces show?
What you will learn
- Represent three coplanar equilibrium forces with a vector triangle
- Use head-to-tail vector addition to show zero resultant
- Resolve or calculate an unknown force from a closed triangle
Equilibrium forces close a vector triangle
For three coplanar forces in equilibrium, their vector sum is zero. Draw each force to scale, head to tail and in its direction. The final arrow returns to the starting point, so the vectors form a closed triangle.
A closed force triangle is a visual statement that the components balance. If the triangle does not close, there is a non-zero resultant force. Use a clear arrow direction and label each force rather than comparing magnitudes alone.
Use components or triangle geometry to find an unknown
Resolve two forces into perpendicular components when convenient. The third force must be equal and opposite to their vector resultant. For perpendicular components, use Pythagoras for magnitude and a tangent ratio for direction.
For a non-right triangle, a scale drawing or appropriate triangle geometry can be used. Keep the chosen axes and angle references explicit, and do not confuse a vector's direction with the direction of the force exerted on a different object.
Pause and explain
A resultant force is 10 N northeast. What force balances it?
Worked example
Two forces on a ring are 6.0 N east and 8.0 N north. Find the third force needed for equilibrium.
Show the worked solution
- The first two forces have a resultant with components 6.0 N east and 8.0 N north.
- Its magnitude is √(6.0² + 8.0²) = 10 N, directed about 53° north of east.
- The balancing force has equal magnitude in the opposite direction.
Answer 10 N at about 53° south of west. The three vectors close a triangle.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
A ring has forces of 3.0 N east and 4.0 N north. Find the magnitude and direction of a third force that puts it in equilibrium.
Give me a hint
Combine the perpendicular forces, then reverse the resultant vector.
Compare my reasoning
- Resultant components are 3.0 N east and 4.0 N north.
- Magnitude = √(3.0² + 4.0²) = 5.0 N; angle = tan⁻¹(4/3) ≈ 53° north of east.
- The balancing force is equal and opposite.
5.0 N at about 53° south of west.
Look for these in your work
- Perpendicular resultant found
- Magnitude and direction calculated
- Opposite balancing direction stated
A 12 N horizontal force and a 5.0 N vertical force act on an object. What is the magnitude of the third force for equilibrium?
Give me a hint
The two perpendicular forces have a resultant found by Pythagoras.
Compare my reasoning
- Resultant magnitude = √(12² + 5.0²).
- √169 = 13 N.
- The third force has the same magnitude and opposite direction to the resultant.
13 N, opposite to the direction of the combined 12 N and 5.0 N resultant.
Look for these in your work
- Components combined correctly
- Magnitude calculated
- Balancing force direction related to resultant
Describe how to check with a scale drawing whether three coplanar forces on a pin are in equilibrium.
Give me a hint
Use a fixed scale and draw arrows head to tail in the actual force directions.
Compare my reasoning
- Choose a scale, such as 1 cm representing 2 N.
- Draw each force vector to scale, with the next arrow starting at the previous arrowhead.
- Check whether the last arrowhead meets the starting point within drawing accuracy.
If the head-to-tail vectors form a closed triangle, their vector sum is zero within the scale drawing's accuracy; if there is a gap, the gap represents a residual force.
Look for these in your work
- Scale and direction retained
- Head-to-tail method described
- Closure interpreted with a limitation
Common mistakes
- Adding force magnitudes without directions
- Drawing vectors tail to tail instead of head to tail
- Using the resultant in the same direction instead of the opposite direction
What can you explain now?
What does a closed head-to-tail triangle of forces show?
Compare with the explanation
The vector resultant is zero
Returning to the start means the vector sum closes to zero.
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Make it stick
What does a gap in a head-to-tail force diagram represent?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.