S3.5 Potential energy
[A] Introduction
- Potential energy is the energy a system stores as a result of its state, shape or position (Key Point 3.3).
- A potential energy can only be defined for a conservative force.
Examples
Reveal
[B] Conservative forces
Key Point 3.9
If the work done by a force in moving an object between two states is independent of the path taken between those two states, the force is conservative.Necessarily, the total work done by a conservative force in moving an object around a closed path is zero. |
- Examples of conservative forces include:
- Frictional forces, such as static friction, dynamic friction (S2.8) and drag are not conservative: these forces dissipate energy (i.e. remove energy from the system). The work they do depends on the path followed by the force.

Commentary
The definition of a conservative force implies that the work done is recoverable. Eg. consider an object thrown vertically upwards in a gravitational field. On the ascent, all its initial kinetic energy is converted into gravitational potential energy. However on descent this potential energy is all converted back into kinetic energy.
Hint for problem solving
Suppose one is interested in the work done by a conservative force in moving an object between two points along some complicated path for which the calculation is difficult. One can replace the actual path taken by some imaginary path for which the calculation is easier. The answer will be the same because for a conservative force the work is independent of the path.

Worked example
Consider a particle moving from point A to point B on a semicircular track of radius R, as shown. What is the work done by gravity on the particle? |
Solution:
The force due to gravity is Fg=-mg and acts vertically downwards at all points. From the definition of work (Key Point 3.4):
![\[ W = \int_{\rm start}^{\rm finish} \vec{F} \cdot d\vec{r} \]](mastermathpng-0.png)
However since gravity is a conservative force, we can (for the purposes of calculating W) replace the actual path by the imaginary path, as shown. Along this path the work done is simply
[C] Potential energy
- Because the work done by a conservative force in moving a body between the two states is unique, we can assign a number to every state, (i.e. a function), that tells us the work done in moving from an arbitrarily chosen reference state to the current state.
- We call this function the potential energy. The work done
in moving between any two states is then the difference in potential
energy of the two states.
Key Point 3.10
Potential Energy. The difference in potential energy, ΔU, between initial and final states is defined as the negative of the work done by the associated force
- Note: The potential energy is often referred to as U rather than ΔU. However, there is always an implied reference state.
[D] Forces from potential energies
- The force acting at any point can be determined from the potential energy.
A convenient way of thinking about forces and potential energies is to imagine walking around a hilly landscape.
Learning Resources
![]() | HRW Chapter 8.1 - 8.3. (HRW 8th Ed has a new section on Reading a P.E. curve, 8.6) |
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