Unit 6 · Topic 05

Torque

Torque measures a force's ability to change rotational motion about an origin or axis.

Unit 6TorqueOpen navigation

Concept 01

Torque direction

Essential The minimum you should retain

Vectorially, τ=r×F \vec\tau=\vec r\times\vec F .

UnderstandInterpret and connect

Its magnitude is rF sinφ and its direction follows the right-hand rule.

DeepenFormulation and conditions

Only the component of F perpendicular to r contributes.

ExploreConnections for further study

Changing the origin changes r and may change torque.

Worked example

Oblique force on a wrench

A 40 N force acts 0.50 m from the axis at φ=30°.

Given
  • F=40 N
  • r=0.50 m
  • φ=30°
Target

Find |τ|.

  1. Model

    τ=rFsinφ.

  2. Substitution

    τ=(0.50)(40)sin30°.

  3. Result

    τ=10 N·m.

  4. Interpretation

    The component parallel to r creates no torque.

Conclusion

|τ|=10 N·m.

Mathematical relation

Torque

τ→=r→×F→,|τ|=rFsinφ=Fℓ \vec\tau=\vec r\times\vec F,\quad |\tau|=rF\sin\phi=F\ell
Represents

Rotational effect of a force.

Physical interpretation

Only the perpendicular component creates torque.

DeepenVariables, conditions, and checks

Variables

ℓ
perpendicular lever arm; usual unit: m
φ
angle between r and F; usual unit: rad or °

Conditions of application

  • Origin or axis is declared.

Dimensional check

N·m, not joules.

Concept 02

Perpendicular lever arm

Essential The minimum you should retain

Lever arm ℓ is the perpendicular distance from the axis to the line of action.

UnderstandInterpret and connect

The same torque can be written τ=Fℓ.

DeepenFormulation and conditions

A large force near the axis may create little torque.

ExploreConnections for further study

A longer wrench increases ℓ without changing F.

A bar, an oblique force, and its line of action show r, φ, and ℓ.Frline of actionℓaxisφτ ⊙

Lever arm ℓ is perpendicular to the line of action and gives |τ|=Fℓ=rFsinφ.

Concept 03

Clockwise and counterclockwise

Essential The minimum you should retain

Planar problems may take counterclockwise positive and clockwise negative.

UnderstandInterpret and connect

Net torque is the algebraic sum of signed contributions.

DeepenFormulation and conditions

The convention must be stated before calculating.

ExploreConnections for further study

The right-hand rule links scalar sign to the axial vector.

Concept 04

N·m is not a joule

Essential The minimum you should retain

Torque is measured in N·m, but the quantity is not called energy.

UnderstandInterpret and connect

Torque and work share dimensions, not physical meaning.

DeepenFormulation and conditions

Every torque needs a stated origin or axis.

ExploreConnections for further study

A force may accelerate the CM and create torque at once.

Concept review

Common errors

Each warning includes a concrete way to review the reasoning, not only an incorrect-answer marker.

Using τ=rF at every angle.

Use τ=rFsinφ=Fℓ.

Calling torque in N·m joules.

Dimensions match, but torque is not energy.