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Three Forces in Equilibrium (Graphical Method)

Key idea: Solve three-force equilibrium problems using the triangle of forces (graphical method).

  • SEC G3 Physics 2027
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Learning objectives

  • Distinguish contact forces from non-contact forces
  • State that mass measures the amount of matter in a body
  • Describe a gravitational field as a region where a mass experiences gravitational force
  • Define gravitational field strength as gravitational force per unit mass
  • Apply weight = mass × gravitational field strength
  • Distinguish mass from weight
  • Describe the effect of balanced and unbalanced forces on a body
  • Describe ways a force may change motion
  • Identify action–reaction pairs on interacting bodies
  • Draw free-body diagrams for force systems in at most two dimensions
  • Solve three-force static equilibrium graphically
  • Apply resultant force = mass × acceleration
  • Relate mass to resistance to change in motion
  • Explain the effects of friction on motion
  • Describe falling with and without air resistance, including terminal velocity
  • Describe a moment as a force's turning effect in everyday examples
  • Apply moment = force × perpendicular distance from the pivot
  • State the principle of moments for a body in equilibrium
  • apply the principle of moments to new situations or to solve related problems
  • show an understanding that the weight of a body may be taken as acting at a single point known as its centre of gravity
  • Explain qualitatively how centre-of-gravity position affects stability

1. Definition

A. Equilibrium (forces)

A point mass is in equilibrium if the resultant force is zero:

∑ vector F = 0

B. Triangle of forces (three forces)

If a point mass is acted on by three forces in equilibrium, the three force vectors drawn head-to-tail form a closed triangle. This is the triangle of forces (graphical method).

What you need for this course

You should be able to solve problems for a static point mass under three forces in two dimensions using a graphical method.

Prerequisites

2. Key Ideas

  • Use an FBD first: Free Body Diagrams
  • If three forces are in equilibrium, the vectors form a closed triangle.
  • Choose a clear scale (e.g. 1 cm : 2 N).
  • Measure unknown magnitude with a ruler and unknown direction with a protractor.
  • State your scale and show a neat, labelled diagram (it earns marks).
Free-body diagram and closed triangle for three-force equilibriumA point has two upward tensions and a downward weight. Beside it, the same three force vectors are drawn head-to-tail as a closed triangle using a stated scale.Three-force equilibriumFree-body diagramHead-to-tail triangleT₁T₂WWT₁T₂same vector scale in both panelsfinish returns to start
Scroll diagram horizontally to read all labels.
For exactly three forces on a static point mass, equilibrium means the scaled head-to-tail vector triangle closes.

3. Detailed Explanations

A. When can you use the triangle of forces?

You can use it when:

  • the object is a point mass (forces act at one point), and
  • there are exactly three forces, and
  • the object is in equilibrium (at rest or moving at constant velocity).

B. Method (step-by-step)

  1. Draw the free body diagram and label the three forces.
  2. Pick a scale (e.g. 1 cm : 2 N).
  3. Draw the first force vector to scale in the correct direction.
  4. From the head of the first vector, draw the second force vector (correct direction, to scale).
  5. From the head of the second, draw the third vector. If equilibrium holds, the triangle should close back to the start point.
  6. If a force is unknown, use the closure of the triangle to determine it by measurement.

4. Common Mistakes

  • Not drawing vectors to scale (then measurements are unreliable).
  • Forgetting that equilibrium means the triangle must close.
  • Using the triangle method when there are more than three forces.
  • Drawing the three forces from the same point (you must draw them head-to-tail for the triangle method).
  • Not stating the scale (or using a scale that gives a tiny, inaccurate diagram).

5. Exam Tips

  • In your answer, state the scale you used.
  • If your triangle does not close, check your directions first.
  • Use long arrows and a sensible scale (bigger diagram = smaller percentage error).
  • Final answers should have units N and (if asked) a direction/angle.

6. Worked Examples

Hidden assumptions to watch for
  • Use the triangle method only when there are exactly three forces and the point mass is in equilibrium.
  • “light string” / “smooth pulley”: tension is the same throughout the string.
  • Your measured value depends on diagram size: a bigger, neater diagram gives a smaller percentage error.

Modelled example 1

Two equal strings (symmetric)

Core

Problem

A 10 N weight is supported symmetrically by two strings, each at 45° above the horizontal. Use a triangle of forces with scale 1 cm : 2 N to find each tension.
Study the worked solution

  1. Draw the known force to scale

    Method

    Draw the 10 N weight as a 5.0 cm vertical arrow downward.

    Reason

    The declared scale converts each 2 N into 1 cm.

    Working

    10 N÷2 N cm⁻¹ = 5.0 cm

  2. Close the force triangle

    Method

    From the weight arrow’s head, draw the two tension directions head-to-tail until the triangle closes.

    Reason

    Three equilibrium forces have zero vector sum.

    Working

    Symmetry makes the two measured tension sides equal, about 3.5 cm each.

  3. Convert the measured length

    Method

    Multiply the measured side by the scale.

    Reason

    A graphical answer must be converted back from centimetres to newtons.

    Working

    T ≈ (3.5 cm)(2 N cm⁻¹) ≈ 7.0 N A neat diagram gives about 7.1 N.

Guided practice 2

Two strings at different angles (non-symmetric)

About 8 min

Problem

A 12 N sign is supported by a left string at 30° above horizontal and a right string at 60° above horizontal. Draw a scaled force triangle and estimate both tensions.

Record measurements from your force triangle

Unit: N
Unit: N

Hints

Hint 1: start with an FBD
Show weight downward and both tensions along their strings.
Hint 2: draw head-to-tail
Use 1 cm : 2 N, draw the 12 N weight as 6.0 cm, then follow the two tension directions to close.
View solution step by step

  1. Construct the triangle

    Method

    Draw the three force directions head-to-tail using a stated scale.

    Reason

    The triangle must close because the sign is in equilibrium.

    Working

    With 1 cm : 2 N, the weight vector is 6.0 cm.

  2. Measure and convert

    Method

    Measure both tension sides and multiply each length by 2 N cm⁻¹.

    Reason

    The unequal angles produce unequal side lengths.

    Working

    T_left ≈ 6 N, T_right ≈ 10.4 N

Common misconception 3

3–4–5 triangle (find the third force exactly)

Find and correct the mistake

Learner response

A point mass has forces 3 N east and 4 N north. A student says the third equilibrium force is 5 N north-east because that is the direction of their resultant. Locate the error.

Choose the closing-vector direction

Third-force direction

View solution step by step

  1. Find the resultant magnitude

    Method

    Use the scaled right-triangle construction, or its 3–4–5 geometry.

    Reason

    The east and north vectors are perpendicular.

    Working

    R = square root of (3² + 4²) = 5 N north-east

  2. Reverse the resultant

    Method

    Draw the third vector back to the starting point.

    Reason

    Equilibrium requires a closed triangle and zero vector sum.

    Working

    The third force is 5 N south-west.

Examiner practice 4

Lamp held by a horizontal pull (three-force equilibrium)

4 marks

Examination question

A lamp at rest has weight 4 N downward, a 3 N horizontal pull left, and an unknown string tension. Use a force triangle to find the tension’s magnitude and general direction. [4 marks]

State scale, construction and result

View solution step by step

  1. Construct a scaled closed triangle

    2 marks

    Method

    Draw the 4 N and 3 N forces head-to-tail using a stated scale, then close the triangle.

    Reason

    The lamp is at rest under exactly three forces.

    Working

    For example, with 1 cm : 1 N, draw sides of 4 cm downward and 3 cm left.

  2. Measure the closing vector

    2 marks

    Method

    Measure its length and read its direction from the force diagram.

    Reason

    The closing vector is the string tension needed for zero resultant.

    Working

    T ≈ 5 N directed upward and to the right.

Challenge 5

5–12–13 triangle (another quick check)

Minimal support

Direction transfer

Two forces on a point mass are 5 N left and 12 N upward. Construct a force triangle and determine the third force needed for equilibrium.

Give magnitude and direction

Hints

Hint 1: draw the known vectors first
Place the leftward and upward forces head-to-tail to scale.
Hint 2: close rather than extend
The third vector runs from the final head back to the original tail.
View solution step by step

  1. Measure the resultant side

    Method

    Use the scaled triangle to measure the diagonal formed by the known forces.

    Reason

    The perpendicular 5 and 12 sides produce a 13-unit diagonal.

    Working

    R = square root of (5² + 12²) = 13 N

  2. Choose the closing direction

    Method

    Reverse the known forces’ resultant.

    Reason

    The third force must close the force triangle.

    Working

    The required force is 13 N downward and to the right.

7. Mind Stretchers

Mind stretcher 1: Find the third force (magnitude and direction)Extension

Two forces act on a point mass:

  • 6 N to the east
  • 8 N to the north

What third force must act so that the point mass is in equilibrium?

Show Answer

In equilibrium, the three forces must form a closed triangle.

Draw the 6 N east and 8 N north head-to-tail, then draw the closing side back to the start.

You should find the third force is about:

  • magnitude ≈ 10 N
  • direction: towards the south-west (opposite of “east then north”)

Mind stretcher 2: Check whether the system can be in equilibriumExtension

Three forces are said to act on a point mass:

  • 3 N, 4 N and 10 N

Can these three forces form a triangle and produce equilibrium? Explain briefly.

Show Answer

No.

For three vectors to form a closed triangle, the largest side must be smaller than the sum of the other two.

But 10 > 3 + 4, so a closed triangle is impossible, and the forces cannot be in equilibrium.

Continue with the next resource in this course.

Course and syllabus information
Course
SEC G3 Physics
Edition
SEC G3 Physics 2027