Classical mechanicsLesson 2 of 5 in the course Potential Energy and Conservation of Energy

Conservative Forces: Path Tests, Potential Energy, and Turning Points

5 flashcards · 5 quiz questions · 5 match pairs · 5 fill-in sentences · English

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Summary

This lesson introduces the difference between conservative and non-conservative forces. The distinction matters because it tells us whether energy changes can be tracked using potential energy alone or whether the path of motion must also be considered.

WAB, path 1=WAB, path 2.W_{AB,\,\text{path 1}}=W_{AB,\,\text{path 2}}.

The route may differ, but the work depends only on the starting and ending positions. Gravity and the ideal spring force are standard examples. For gravity near Earth's surface, the change in gravitational potential energy depends on the change in height, not on the route taken to reach that height.

All flashcards

What is a conservative force?
A force whose work between fixed endpoints is independent of the path.
What does the closed-path test say about a conservative force?
Its total work around any closed path is zero: ∮F⃗⋅dr⃗=0\oint \vec{F}\cdot d\vec{r}=0.
What is the two-dimensional local derivative test for a conservative force field?
For a smooth field, compare ∂Fx/∂y\partial F_x/\partial y with ∂Fy/∂x\partial F_y/\partial x; global conclusions also depend on the domain.
How is a conservative force related to potential energy?
The force is the negative gradient of potential energy: F⃗=−∇U\vec{F}=-\nabla U.
What happens at a turning point when mechanical energy is conserved?
Kinetic energy is zero momentarily, so potential energy equals total energy.

Quiz questions

  1. What makes a force conservative?Answer: Its work between fixed endpoints is independent of the path.
  2. What is the total work of a conservative force around any closed path?Answer: Zero.
  3. For a smooth two-dimensional force field, which equality is the local derivative test for conservativeness?Answer: ∂Fx∂y=∂Fy∂x\frac{\partial F_x}{\partial y}=\frac{\partial F_y}{\partial x}
  4. How is force related to potential energy UU?Answer: Force equals the negative gradient of UU.
  5. Which pair consists of familiar conservative forces?Answer: Gravity and ideal spring force.

Based on University Physics Volume 1 by William Moebs, Samuel J. Ling, Jeff Sanny (OpenStax), licensed CC BY 4.0. Adapted into a summary, flashcards and quizzes.

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