Work Done by a Variable Force
Calculation of work done when the applied force varies with position, using graphical methods and integration.
What is Work Done by a Variable Force?
A force whose magnitude or direction, or both, change with time or position.
Key formula / rule: Work done by variable force (1D)
Key points
- Define work done by a variable force.
- Calculate work done using integration of F(x) dx.
- Determine work done from the area under an F-x graph.
- Relate the concept to real-world phenomena.
Common exam trap
Confusing variable force integration with constant force calculation (FΔx).
Definitions
- Term
Variable Force
- Meaning
A force whose magnitude or direction, or both, change with time or position.
- Term
Work Done
- Meaning
The energy transferred to or from an object via the application of force along a displacement. For a variable force, it's the integral of force over displacement.
Learning objectives
Define work done by a variable force.
Calculate work done using integration of F(x) dx.
Determine work done from the area under an F-x graph.
Relate the concept to real-world phenomena.
Formulae
- Name
Work done by variable force (1D)
- Note
F(x) is the force as a function of position x, and x₁ and x₂ are the initial and final positions.
- Expression
W = \intx_1^{x2} F(x) dx
- Name
Work done by variable force (3D)
- Note
Line integral of force vector \vec{F} along path C.
- Expression
W = \ointC \vec{F} \· d\vec{r}
- Name
Work done by constant force
- Note
This is a special case of the variable force formula where F is constant.
- Expression
W = F \Δ x
Prerequisites
Understanding of basic work definition (W = Fd cosθ).
Familiarity with integration in calculus.
Ability to interpret graphs (area under the curve).
Common mistakes
Confusing variable force integration with constant force calculation (FΔx).
Incorrectly calculating the area under the F-x graph (e.g., missing parts of the area or using wrong geometric formulas).
Errors in setting up the integration limits (x₁ and x₂).
Not considering the sign of the force or displacement correctly in the integral.
Keywords
Variable Force
Work Done
Integration
Force-Displacement Graph
Calculus
Area Under Curve
Practice preview
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