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Equilibrium, Rotational Equilibrium & Translational Equilibrium

Key idea: Learn the conditions for equilibrium (resultant force = 0 and resultant moment = 0) and how to apply the principle of moments (O Level Physics 6091).

  • 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 (rigid body)

A rigid body is in equilibrium if:

  • the resultant force is zero (no linear acceleration), and
  • the resultant moment about any point is zero (no angular acceleration).

B. Translational equilibrium

Translational equilibrium means the resultant force is zero:

Fᵣₑₛᵤₗₜₐₙₜ = 0

C. Rotational equilibrium (principle of moments)

Rotational equilibrium means the total clockwise moment equals the total anticlockwise moment about the same pivot. Equivalently:

∑ M = 0

2. Key Ideas

  • Use both equilibrium conditions for a rigid body:
    • Fᵣₑₛᵤₗₜₐₙₜ = 0
    • ∑ M = 0 (sum of moments is zero)
  • Moment of a force about a pivot:
    • M = F × d
    • d is the perpendicular distance from the pivot to the line of action (see Moment Of A Force).
  • Units:
    • force in N
    • distance in m
    • moment in N m
  • When an object is in equilibrium, it can be:
    • at rest, or
    • moving at constant velocity (but most turning-effect questions are static equilibrium).

Balancing moments: heavier force needs smaller distance

Distance needed for balance versus weight, for a fixed moment (W × d constant), showing an inverse relationship.

Scroll across the graph to read all labels.

Distance needed for balance versus weight, for a fixed moment (W × d constant), showing an inverse relationship.Distance needed for balance versus weight, for a fixed moment (W × d constant), showing an inverse relationship.
Example: balancing a fixed moment of 240 N m (e.g. 400 N at 0.60 m). In equilibrium, clockwise moment = anticlockwise moment, so W is inversely related to distance.
Open full-size graph
View figure data
Values for Balancing moments: heavier force needs smaller distance
Weight on other side (N)W d = constant
2001.2
2401
3000.8
4000.6
6000.4

3. Detailed Explanations

Balanced clockwise and anticlockwise momentsA beam pivots on a triangular support. A 120-newton downward force acts 6 metres left of the pivot, and force F acts downward 8 metres to the right.pivot120 NF6 m8 manticlockwise momentclockwise moment
In rotational equilibrium: total clockwise moment = total anticlockwise moment about the pivot.

A. Two conditions, two types of equilibrium

  1. Translational equilibrium (no change in velocity): resultant force is zero.
  2. Rotational equilibrium (no turning): resultant moment is zero.

For a rigid body to be “fully balanced”, it must satisfy both.

B. How to solve equilibrium questions (O Level workflow)

  1. Draw a clear diagram and mark the pivot.
  2. Draw an FBD if needed: Free Body Diagrams (FBD).
  3. Label forces and perpendicular distances (convert cm → m).
  4. Use the principle of moments:
    • total clockwise moments = total anticlockwise moments
  5. Use force balance if needed:
    • up forces = down forces (vertical equilibrium)
    • left forces = right forces (horizontal equilibrium)
Sign convention (moments)

You can use either method:

  • no signs: total clockwise moments = total anticlockwise moments, or
  • with signs: choose clockwise positive, then write ∑ M = 0.

4. Common Mistakes

  • Using the distance along a rod instead of the perpendicular distance to the force’s line of action.
  • Forgetting to convert cm to m, giving moments in the wrong unit.
  • Taking moments about the wrong pivot (or changing pivot mid-solution).
  • Using mass (kg) instead of weight (N) when the force is due to gravity.
  • Using the principle of moments when the object is not in equilibrium.

5. Exam Tips

  • Pick the pivot to simplify the math (often choose the pivot where an unknown force acts, so its moment is zero).
  • Write the equation in words first:
    • “clockwise moments = anticlockwise moments”
  • Show units clearly and give final answers in N or N m.
  • If the question says “balanced” or “level” or “not rotating”, that is a strong hint to use the principle of moments.

6. Worked Examples

Hidden assumptions to watch for
  • Equilibrium means both: Fᵣₑₛᵤₗₜₐₙₜ = 0 and ∑ M = 0.
  • “uniform beam/rule”: weight acts at the centre.
  • Use perpendicular distances and convert cm → m for moments in N m.
  • Use the value of g given (or 10 N kg⁻¹ if not stated).

Modelled example 1

Balanced beam (find an unknown force)

Core

Problem

In a diagram, a beam is supported at a pivot.

  • A downward force of 120 N acts 6.0 m to the left of the pivot.
  • A downward force F acts 8.0 m to the right of the pivot.

The beam is balanced. Find F.

Study the worked solution

  1. Choose the pivot and turning directions

    Method

    Take moments about the support and separate the anticlockwise and clockwise effects.

    Reason

    The support force then has zero moment, and equilibrium requires the opposing moments to balance.

    Working

    The 120 N force acts anticlockwise; F acts clockwise.

  2. Calculate the known moment

    Reason

    The 6.0 m distance is perpendicular to the vertical force’s line of action.

    Working

    M_ACW = (120)(6.0) = 720 N m

  3. Balance the moments and solve

    Reason

    A balanced beam has zero resultant moment about the pivot.

    Working

    720 = 8.0F ⇒ F = 90 N

Guided practice 2

Find the balancing distance (principle of moments)

About 4 min

Problem

A boy of weight 400 N sits 0.60 m from the pivot of a seesaw. A girl of weight 300 N sits on the other side.

How far from the pivot should the girl sit for the seesaw to balance?

Complete the balancing calculation

Unit: m

Hints

Hint 1: identify what equilibrium requires
The two people must produce equal moments in opposite directions.
Hint 2: set up the moment equation
Write 400(0.60) = 300d before solving for d.
View solution step by step

  1. Set up the equilibrium condition

    Method

    Equate the clockwise and anticlockwise moments about the pivot.

    Reason

    The seesaw is balanced, so its resultant moment is zero.

    Working

    400(0.60) = 300d

  2. Solve for the distance

    Reason

    Dividing the boy’s moment by the girl’s weight gives the moment arm she needs.

    Working

    d = (400 × 0.60)/300 = 0.80 m

Common misconception 3

Translational equilibrium is not the only condition

Find and correct the mistake

Learner response

A book rests on a table. Its normal contact force N is upwards and its weight W is downwards along the same vertical line. A student writes:

The student’s claim is: Because N = W, checking the resultant force is always enough to prove that a rigid body is fully in equilibrium.

Locate the first error and correct the argument for this book.

Diagnose the argument before viewing the correction

What is the first error?

View solution step by step

  1. Locate the first error

    Method

    Reject the claim that force balance is always sufficient.

    Reason

    A rigid body can have zero resultant force but a non-zero resultant moment if equal opposing forces act along different lines.

    Working

    Full equilibrium requires both Fᵣₑₛᵤₗₜₐₙₜ = 0 and ∑ M = 0.

  2. Correct the argument for this book

    Reason

    Here N and W are equal, opposite and act along the same vertical line, so neither a resultant force nor a resultant moment remains.

    Working

    N-W = 0, ∑ M = 0

Examiner practice 4

Supported at both ends (use moments and force balance)

5 marks

Examination question

A 2.0 m uniform couch is supported upward at its left and right ends by forces F_L and F_R. A 60 kg person's weight acts downward 0.50 m from the left end. The 20 kg couch's weight acts downward at its centre, 1.0 m from the left end.
Use moments about one end to find the other support force, then use vertical force balance.

A 2.0 m couch of mass 20 kg is carried level by two people lifting at the ends. A person of mass 60 kg sits 0.50 m from the left end. The couch’s centre of gravity is at the middle.

Take g = 10 N kg⁻¹. Calculate the upward force at each end. [5 marks]

Write your solution before viewing the mark scheme

View solution step by step

  1. Convert both masses to weights

    1 mark

    Method

    Find the two downward forces and locate their lines of action.

    Reason

    Moments and force balance use forces in newtons, not masses in kilograms.

    Working

    W_c = (20)(10) = 200 N at 1.0 m; Wₚ = (60)(10) = 600 N at 0.50 m from the left end.

  2. Write the moment equation about the left end

    1 mark

    Method

    Take moments about the left support.

    Reason

    The unknown left support force then has zero moment, leaving one support force to determine.

    Working

    F_R(2.0) = (600)(0.50) + (200)(1.0)

  3. Find the right support force

    1 mark

    Reason

    The couch is rotationally balanced, so the clockwise and anticlockwise moments are equal.

    Working

    F_R = (300 + 200)/2.0 = 250 N

  4. Apply vertical force balance

    1 mark

    Reason

    The couch has no vertical acceleration, so total upward force equals total downward force.

    Working

    F_L + F_R = 600 + 200 = 800 N

  5. Find the left support force

    1 mark

    Working

    F_L = 800-250 = 550 N

Challenge 5

Seesaw balancing (different weights)

Minimal support

Symbolic transfer

Two people sit on opposite sides of a seesaw. Person A has weight W_A and person B has weight W_B, where W_A > W_B.

Without choosing numerical values, explain how they can balance the seesaw and compare their distances d_A and d_B from the pivot.

Commit to the comparison before viewing the reasoning

Which distance comparison allows balance?

Hints

Hint 1: write the symbolic balance condition
For balance, write one weight-distance product for each side.
Hint 2: compare the distances
If W_A is larger, its distance must be smaller for the products to remain equal.
View solution step by step

  1. Express the balance symbolically

    Method

    Equate the opposing moments about the pivot.

    Reason

    A balanced seesaw has zero resultant moment even when the two weights differ.

    Working

    W_A d_A = W_B d_B

  2. Infer the distance comparison

    Reason

    Since W_A > W_B, equality is possible only if A’s moment arm is the smaller one.

    Working

    d_A < d_B

  3. State the physical arrangement

    Working

    The heavier person A sits closer to the pivot, while the lighter person B sits farther away.

7. Mind Stretchers

Mind stretcher 1: Choosing the pivot to eliminate an unknownExtension

A uniform beam is supported at a pivot. There is an unknown reaction force at the pivot.

Why is it often best to take moments about the pivot?

Show Answer

If you take moments about the pivot, the perpendicular distance for the pivot’s reaction force is zero.

So its moment is zero and it disappears from the moment equation, making the calculation simpler.

Mind stretcher 2: When “balanced” does not mean “no forces”Extension

A sign is hanging from two wires and is not moving.

Explain why the forces are not zero even though the sign is in equilibrium.

Show Answer

The sign is in equilibrium because the resultant force is zero, not because all forces are zero.

The weight acts downwards, and the tensions in the wires act upwards. These forces balance so the vector sum is zero.

Continue with the next resource in this course.

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