Conservative Forces and the Potential Energy Function
A force whose work depends only on the endpoints defines a potential energy, and the force is recovered as minus the derivative of that function. Friction fails the test, which is why no friction potential exists.
What a learner can do afterwards
- Tests whether a stated force is conservative by comparing work along different paths
- Builds the potential energy function for gravity and for a spring by integrating the force
- Recovers a force from a potential energy function by differentiating
1 · Read
Some forces bank every joule you put in, so you can draw it back later. Gravity works this way: lifting a ball stores energy, and falling returns it. Compare the work along two different paths between the same points. Equal work means the force is conservative.
There is a second test with the same meaning. Carry something around any closed loop back to the start. A conservative force totals zero work around the loop, because what one stretch takes, the return gives back. Friction fails this test, since every extra lap costs more energy.
Lifting against gravity stores mass times g times height, and the work you did is banked as gravitational potential. Stretching or squeezing a spring banks half k times stretch squared, the same either way. Potential difference is defined as minus the work, with the zero level free for you to choose.
You can recover the force back from its potential by differentiating and flipping the sign. Minus the slope of the potential gives the force, so the force points toward lower potential. That is why friction and drag own no potential: their work depends on the path, so no single function can store it.
Conservative forces bank path independent work as potential, and the force is minus the slope of that potential.
The rewritten readings were written for this dataset.
2 · Watch
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Where it sits
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Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.