Equilibrium, Moments, and Couples

Overview

Equilibrium, Moments, and Couples develops the statics part of Forces. It explains how objects remain balanced translationally and rotationally.

This page focuses on:

• moment of a force
• turning effects and rotational sense
• couple
• torque of a couple
• principle of moments
• conditions for equilibrium
• centre of gravity
• stability and toppling
• choosing pivots
• worked examples
• exam pitfalls

These ideas are heavily tested in beam, ladder, hinge, support, and balance problems.

Why It Matters

Static equilibrium problems require both force balance and moment balance. A body can have zero resultant force but still rotate if the resultant moment is not zero.

Definition

A body is in static equilibrium when it has no translational acceleration and no angular acceleration.

Key Representations

$$\sum \vec{F}=0$$ $$\sum F_x=0,\sum F_y=0$$ $$M=Fd$$ $$\sum M=0$$ $$\tau =Fd$$

What is Equilibrium?

A body is in equilibrium when it has:

• no linear acceleration
• no angular acceleration

For a rigid body, two conditions must both be satisfied.

Translational Equilibrium

$$\sum \vec{F}=0$$

Rotational Equilibrium

$$\sum M=0$$

where the sum of clockwise moments equals the sum of anticlockwise moments about the same pivot.

Force is a Vector

Force has magnitude and direction.

Examples:

• $\vec{W}$ weight
• $\vec{N}$ normal force
• $\vec{T}$ tension

In one dimension, signed scalar components may be used after choosing a positive direction.

Moment of a Force

A moment is the turning effect of a force about a point or pivot.

Magnitude:

$$M=Fd$$

where:

• $F$ = magnitude of force
• $d$ = perpendicular distance from pivot to line of action of force

SI unit:

$$\text{N m}$$

Direction / Sense of a Moment

Moments may be described as:

• clockwise
• anticlockwise

If using sign convention, define clearly, e.g.

• anticlockwise positive
• clockwise negative

Then rotational equilibrium may be written:

$$\sum M=0$$

Perpendicular Distance Matters

Use the shortest perpendicular distance from pivot to the line of action of the force.

Not the slanted distance to point of application.

This is one of the most common mistakes.

Example of Moment

A $10\text{N}$ force acts perpendicular to a spanner $0.25\text{m}$ from pivot.

$$M=Fd=10(0.25)=2.5\text{N m}$$

Couple

A couple consists of:

• two equal forces
• opposite in direction
• parallel
• separated by distance
• lines of action do not coincide

A couple causes rotation only.

No resultant force.

Torque of a Couple

Magnitude:

$$\tau =Fd$$

where:

• $F$ = one of the forces
• $d$ = perpendicular distance between lines of action

Key Difference from Single Force

A single force may cause translation and rotation.

A pure couple causes rotation only.

Principle of Moments

For a body in equilibrium:

Sum of clockwise moments = Sum of anticlockwise moments

Equivalent form:

$$\sum M=0$$

about any chosen pivot.

This is often the fastest way to solve support-force problems.

Choosing a Pivot

Choose a pivot through an unknown force when possible.

Why?

That force then has zero moment arm and drops out of the moment equation.

This simplifies calculations greatly.

Conditions for Equilibrium

Full Rigid-Body Equilibrium

Use both:

$$\sum \vec{F}=0$$

and

$$\sum M=0$$

In Components

For 2D systems:

$$\sum F_x=0$$ $$\sum F_y=0$$ $$\sum M=0$$

Centre of Gravity

The centre of gravity (C.G.) is the point through which the weight of a body may be considered to act.

For uniform symmetrical bodies, it lies at the geometric centre.

Useful in:

• beam problems
• balancing
• stability
• toppling

Stability

A body is stable if small displacement does not cause it to topple easily.

Greater Stability When:

• lower centre of gravity
• wider base area

Toppling Condition

The body topples when the line of action of weight passes outside the base.

Uniform Beam Ideas

For a uniform beam:

• weight acts at midpoint
• if mass of beam is significant, include beam weight
• support reactions may act at ends / hinges

Worked Examples

Example 1: Balanced Beam

A beam is pivoted at centre.

A $20\text{N}$ force acts $0.30\text{m}$ left of pivot.

Find force $F$ needed $0.15\text{m}$ right of pivot.

Clockwise = Anticlockwise:

$$20(0.30)=F(0.15)$$ $$F=40\text{N}$$

Example 2: Ladder / Rod Support

A uniform rod weight $60\text{N}$ is supported at both ends.

Symmetry gives equal reactions:

$$R_A=R_B=30\text{N}$$

Example 3: Couple

Two opposite parallel forces $12\text{N}$ separated by $0.40\text{m}$.

$$\tau =Fd=12(0.40)=4.8\text{N m}$$

Example 4: Force Balance

A signboard hangs at rest.

Then:

$$\sum \vec{F}=0$$

and

$$\sum M=0$$

Both must be satisfied.

Typical Procedure for Statics Problems

1. Draw free-body diagram.
2. Label all forces.
3. Choose axes.
4. Apply:

$$\sum F_x=0,\sum F_y=0$$

5. Take moments about convenient pivot.
6. Solve unknowns.
7. Check units and reasonableness.

Common Exam Pitfalls

1. Using wrong distance in moment

Use perpendicular distance to line of action.

2. Forgetting beam weight

Uniform beam weight acts at midpoint.

3. Using only force balance

Rigid body equilibrium usually needs moment balance too.

4. Wrong turning direction signs

Choose convention and stay consistent.

5. Calling a single force a couple

A couple needs two equal opposite separated forces.

6. Forgetting reaction forces at supports

Hinges and supports often exert unknown reactions.

7. Assuming centre of gravity always at geometric centre

Only true for uniform symmetrical bodies.

Summary

Core Equations

Force Equilibrium

$$\sum \vec{F}=0$$

Component Forms

$$\sum F_x=0,\sum F_y=0$$

Moment of a Force

$$M=Fd$$

Rotational Equilibrium

$$\sum M=0$$

Torque of a Couple

$$\tau =Fd$$

Big Ideas

• forces balance translation
• moments balance rotation
• couples rotate without translation
• choose pivots strategically
• stability depends on base and C.G.

Related Links

• Forces
• Force Diagrams and Resolution
• Fluid Forces and Resistive Motion
• Dynamics
• Kinematics
• Vectors

Links

• Forces
• Force Diagrams and Resolution
• Dynamics
• Vectors