Unit 4 · Topic 01
Work by a force
Mechanical work describes an energy transfer associated with a force acting through a displacement. Its sign and value depend on the force component along the motion, not on everyday effort.
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Concept 01
Transfer by a force
Essential The minimum you should retain
A force does work when it acts while its point of application undergoes a displacement.
UnderstandInterpret and connect
The mere presence of a force is not enough: its component along the displacement matters.
DeepenFormulation and conditions
The work of a constant force is the dot product W = F·Δr.
ExploreConnections for further study
Holding an object still may require physiological effort, but it does zero mechanical work on the object because there is no displacement.
Worked example
Pulling with an angled force
A box moves 6.0 m horizontally while a constant 80 N force acts 30° above horizontal.
- F=80 N
- Δr=6.0 m
- θ=30°
Calculate only the work by the applied force.
- Principle
Use W=FΔr cosθ.
- Representation
θ is the angle between force and displacement.
- Calculation
W=(80)(6.0)cos30°≈416 J.
- Interpretation
Work is positive; the vertical component does not contribute.
Mathematical relation
Work by a constant force
Work by a constant force through a net displacement.
Only the parallel force component transfers energy through work.
DeepenVariables, conditions, and checks
Variables
- W
- work done by the force; usual unit: J
- F
- force magnitude; usual unit: N
- Δr
- displacement magnitude; usual unit: m
- θ
- angle between force and displacement; usual unit: rad or °
Conditions of application
- The force is constant.
- θ is measured between F and Δr.
Dimensional check
N·m=J.
Errors it helps prevent
- Using W=FΔr without considering the angle.
The F cosθ component aligns with Δr; the perpendicular component does not contribute to the dot product.
Concept 02
Angle and parallel component
Essential The minimum you should retain
If F makes an angle θ with Δr, only F cosθ contributes to work.
UnderstandInterpret and connect
The perpendicular component can change the direction of motion without changing kinetic energy.
DeepenFormulation and conditions
For a constant force, W = F Δr cosθ, where θ is the angle between the vectors.
ExploreConnections for further study
In uniform circular motion a radial resultant is perpendicular to instantaneous displacement and does zero net work.
The sign follows from comparing F with displacement, not from everyday effort.
Concept 03
The sign has physical meaning
Essential The minimum you should retain
Positive, zero, or negative work describes how a force transfers energy during displacement.
UnderstandInterpret and connect
Angles below 90° give W>0, 90° gives W=0, and angles above 90° give W<0.
DeepenFormulation and conditions
Work is a scalar and its SI unit is the joule: 1 J = 1 N·m.
ExploreConnections for further study
Zero work does not require zero force; it can also result from perpendicularity or cancellation.
Concept 04
Each force contributes work
Essential The minimum you should retain
Different forces can do different amounts of work over the same displacement.
UnderstandInterpret and connect
Their contributions may have different signs and should remain separate before summing.
DeepenFormulation and conditions
Net work is the scalar sum of the work done by every force.
ExploreConnections for further study
There is no universal rule that normal force or tension must always do zero work; geometry and the point of application matter.
Concept review
Common errors
Each warning includes a concrete way to review the reasoning, not only an incorrect-answer marker.
Confusing human effort with mechanical work.
Work on the body depends on force and displacement; holding it still gives W=0 on it.
Using W=FΔr without the angle.
In general W=FΔr cosθ; omitting cosine requires parallel force in the same direction.