Conservative Forces and Potential Energy
A force is conservative when the work it does is path independent; the associated potential energy satisfies F = -dU/dx.
What is Conservative Forces and Potential Energy?
A force for which work done is independent of the path and depends only on the start and end points.
Key formula / rule: Force-Potential Relationship (1D)
Key points
- Distinguish between conservative and non-conservative forces.
- Calculate potential energy for given force functions.
- Determine force components using partial derivatives of U.
- Identify equilibrium types from potential energy curves.
Common exam trap
Forgetting the negative sign in F = -dU/dx.
Definitions
- Term
Conservative Force
- Meaning
A force for which work done is independent of the path and depends only on the start and end points.
- Term
Potential Energy
- Meaning
The energy possessed by a body by virtue of its position or configuration in a conservative force field.
Learning objectives
Distinguish between conservative and non-conservative forces.
Calculate potential energy for given force functions.
Determine force components using partial derivatives of U.
Identify equilibrium types from potential energy curves.
Formulae
- Name
Force-Potential Relationship (1D)
- Note
Force is the negative derivative of potential energy.
- Expression
F = -dU/dx
- Name
Force-Potential Relationship (3D)
- Note
Force is the negative gradient of U.
- Expression
F = - ( (∂U/∂x)i + (∂U/∂y)j + (∂U/∂z)k )
- Name
Change in Potential Energy
- Note
Integration from initial to final position.
- Expression
ΔU = - ∫ F·dr
Prerequisites
Work-Energy Theorem
Vector Calculus (basics of gradients)
Definite Integration
Common mistakes
Forgetting the negative sign in F = -dU/dx.
Applying potential energy concepts to friction or air resistance.
Confusing total energy with potential energy on a graph.
Keywords
Conservative Force
Potential Energy
Gradient
Path Independence
Mechanical Energy
Practice preview
Which of the following is an example of a conservative force?…
easy
If the potential energy U(x) of a particle is given by U(x) = (1/2)kx^2, where k is a positive constant, what is the conservative force F(x) acting on the particle?…
easy
A particle is moved from point A to point B and then back to point A along a different path, under the action of a conservative force. What is the total work done by the conservative force during this closed loop?…
medium