Work Done by Constant and Variable Forces
Work is the dot product of force and displacement, W = F.s = Fs cos t, generalised to W = integral F.dx for a variable force.
What is Work Done by Constant and Variable Forces?
The amount of work done when a force of one Newton displaces a body by one meter in the direction of the force.
Key formula / rule: Work Done (Constant Force)
Key points
- Differentiate between work done by constant and variable forces.
- Calculate work using vector dot products and integration.
- Interpret Force-Displacement graphs to calculate energy transfer.
Common exam trap
Using W=Fs directly for variable forces without integration.
Definitions
- Term
Joule
- Meaning
The amount of work done when a force of one Newton displaces a body by one meter in the direction of the force.
- Term
Conservative Force
- Meaning
A force for which the work done in moving a particle between two points is independent of the path taken.
Learning objectives
Differentiate between work done by constant and variable forces.
Calculate work using vector dot products and integration.
Interpret Force-Displacement graphs to calculate energy transfer.
Formulae
- Name
Work Done (Constant Force)
- Note
θ is the angle between Force and Displacement.
- Expression
W = F · s = Fs \cos \θ
- Name
Work Done (Variable Force)
- Note
Evaluated as the area under the F vs x graph.
- Expression
W = \int_{x1}^{x2} F(x) dx
- Name
Work in Vector Form
- Note
Used for 3D force fields.
- Expression
W = \int (Fx dx + Fy dy + Fz dz)
Prerequisites
Vector Algebra (Dot Product)
Basic Differential and Integral Calculus
Newton's Laws of Motion
Common mistakes
Using W=Fs directly for variable forces without integration.
Incorrectly identifying the angle θ between F and s.
Forgetting to include the sign (positive/negative) for work done by friction.
Confusing work with power (rate of doing work).
Keywords
Scalar Product
Integration
Joule
Conservative Force
Dot Product
Practice preview
What is the work done by a constant force F acting on an object that undergoes a displacement s?…
easy
A porter lifts a suitcase and walks horizontally on a platform. What is the work done by the gravitational force on the suitcase?…
easy
A force of 10 N acts on an object at an angle of 60 degrees with the horizontal. If the object is displaced by 5 m horizontally, what is the work done by the force?…
medium