The Work-Energy Theorem for a Variable Force
Extension of the work-energy theorem to cases where forces are variable, involving integral forms of work.
What is The Work-Energy Theorem for a Variable Force?
A force whose magnitude or direction (or both) changes as the object moves or with time.
Key formula / rule: Work done by a variable force
Key points
- To calculate work done by a variable force using integration.
- To apply the Work-Energy Theorem for variable forces.
- To relate the net work done to the change in kinetic energy in variable force scenarios.
Common exam trap
Confusing work done by a specific force with net work done.
Definitions
- Term
Variable Force
- Meaning
A force whose magnitude or direction (or both) changes as the object moves or with time.
- Term
Net Work
- Meaning
The sum of the work done by all individual forces acting on an object.
- Term
Kinetic Energy
- Meaning
The energy possessed by an object due to its motion.
Learning objectives
To calculate work done by a variable force using integration.
To apply the Work-Energy Theorem for variable forces.
To relate the net work done to the change in kinetic energy in variable force scenarios.
Formulae
- Name
Work done by a variable force
- Note
This formula is used when the force F is a function of position x.
- Expression
W = ∫[x₁ to x₂] F(x) dx
- Name
Work-Energy Theorem
- Note
The net work done on an object equals the change in its kinetic energy.
- Expression
Wnet = ΔK = Kf - Ki
- Name
Kinetic Energy
- Note
Where m is mass and v is velocity.
- Expression
K = (1/2)mv²
Prerequisites
Understanding of work done by a constant force.
Basic concepts of kinetic energy and its formula.
Familiarity with integral calculus (definite integrals).
Common mistakes
Confusing work done by a specific force with net work done.
Incorrectly applying the formula for constant force to variable force situations.
Errors in setting up or evaluating the definite integral.
Forgetting to consider all forces contributing to the net work.
Keywords
Work-Energy Theorem
Variable Force
Integration
Kinetic Energy
Work
Displacement
Calculus in Physics
Practice preview
How is the work done by a variable force F on an object moving from position r1 to r2 calculated?…
easy
According to the Work-Energy Theorem, the net work done by all forces acting on an object is equal to:…
easy
A particle of mass 2 kg moves along the x-axis under the action of a variable force F = (4x - 2) N. If the particle starts from rest at x = 0, what is its kinetic energy when it reaches x = 3 m?…
medium