Moments, couples and centre of gravity

Key idea: A force turns a body only through its perpendicular lever arm. A couple produces rotation without a resultant force, while centre of gravity locates the line of action of weight.

GCE A-Level H1 Physics 2027

H1 Physics 8867 · Lesson 2 of 3

Check your understanding

By the end of this lesson, you should be able to

Define and calculate the moment of a force and torque of a couple.
Explain why a couple produces rotation only.
Use the centre of gravity as the point through which a body’s weight acts.

Learn the idea

Big question: Why can the same force produce a large turn, a small turn or no turn at all?

The line of action controls turning

Moment is not determined by the distance to the point where the force arrow begins. It uses the shortest distance from the pivot to the force’s complete line of action. If that line passes through the pivot, the moment is zero even when the force acts far from the pivot on the drawing.

Check your understanding: Why does pushing directly towards a door hinge produce almost no turning?

The force’s line of action passes close to the hinge, so its perpendicular moment arm is nearly zero.

A couple turns without translating

The equal and opposite forces of a couple cancel as a resultant force, yet both moments have the same rotational sense. Its torque Fd uses the separation between the two parallel lines of action and does not depend on the chosen pivot.

Centre of gravity lets a distributed weight be replaced by one resultant force for force and moment calculations. This is why the weight of a uniform beam can be drawn at its midpoint.

Check your understanding: If the separation of a couple doubles at the same force, what happens to its torque?

It doubles because couple torque is Fd.

Moment uses the perpendicular distance from the pivot to the force’s line of action. Weight acts through a body’s centre of gravity.

Key ideas

Moment depends on the chosen pivot.
Use the shortest distance to the line of action.
A zero resultant force does not imply zero torque.

Relationships to know

moment = Fd⊥
couple torque = F × perpendicular force separation

Follow the reasoning

Worked example

Moment of an angled force

Question: A 30 N force acts at the end of a 0.50 m handle and makes 40° with the handle. Find its moment about the pivot.

Step 1: Identify the turning component
Why: Only the component perpendicular to the handle contributes to the moment.
Working: F⊥ = 30 sin40° = 19.3 N.
Step 2: Multiply by the radius
Why: The perpendicular component acts 0.50 m from the pivot.
Working: M = F⊥r = 19.3(0.50) = 9.64 N m.
Step 3: State the rotational sense
Why: Moment is a directed turning effect in a plane.
Working: Use the diagram to label the result clockwise or anticlockwise.

Answer: The moment magnitude is 9.6 N m, with the sense determined by the force arrow in the diagram.

Check: The value is below the maximum 30(0.50) = 15 N m because the force is not perpendicular.

Now try it with support

Practise with support

A 30 N force acts 0.50 m from a pivot at 40° to the radius. Find its moment magnitude.

Hints

Only the component perpendicular to the radius turns the body.
Use F r sin θ.

View the guided answer

M = 30(0.50) sin 40° = 9.64 N m.

Your turn

Practise independently

Two equal 18 N forces form a couple with perpendicular separation 0.40 m. Find the torque and explain why the resultant force is zero.

Check your answer

The torque is one force times the perpendicular separation: τ = 18(0.40) = 7.2 N m. The forces cancel translationally because they are equal and opposite, but both moments turn in the same sense.

Common mistakes and exam guidance

Watch out for

Using the distance to the point of application instead of the perpendicular distance.
Doubling Fd for a couple when d is already the separation between the two force lines.

In an exam

State clockwise or anticlockwise when the direction is requested.
Place the weight of a uniform beam at its midpoint before taking moments.

Put the ideas together

Exam-style practice [6 marks]

A uniform 4.0 m beam of weight 120 N is pivoted 1.0 m from its left end. An 80 N load acts at the left end and an upward force F acts at the right end. Find F and the vertical pivot force for equilibrium.

Plan before you answer

Place the beam’s weight at its midpoint.
Take moments about the pivot.
Use vertical force balance after finding F.

View the marking points and model answer

Marking points

Locates beam weight 1.0 m right of pivot.
Uses 80 N load 1.0 m left of pivot.
Uses F with 3.0 m arm.
Writes 80(1.0) + 3.0F = 120(1.0).
Obtains F = 13.3 N upward.
Obtains pivot force 186.7 N upward.

Model answer

About the pivot, the 80 N load and F act anticlockwise while the beam’s weight acts clockwise: 80(1.0) + F(3.0) = 120(1.0). Thus F = 13.3 N upward. Vertical balance gives Rpivot + 13.3 = 120 + 80, so Rpivot = 186.7 N upward.

Finish from memory

Three-question recap

Define the moment of a force about a point.
Check
Force multiplied by the perpendicular distance from the point to the force’s line of action.
Why does a couple have zero resultant force?
Check
Its two forces are equal in magnitude and opposite in direction.
Where does the weight of a uniform beam act in the model?
Check
Through its centre of gravity at the midpoint.

Try this next

Use moments and centre of gravity together in a supported-beam equilibrium problem.

Continue to Translational and rotational equilibrium Return to Forces and Moments Review Field, contact and elastic forces

Continue with the next resource in this course.

Course and syllabus information

Course: GCE A-Level H1 Physics
Edition: GCE A-Level H1 Physics 2027