General physicsLesson 4 of 5 in the course Potential Energy and Conservation of Energy

Potential-Energy Diagrams: Allowed Motion, Equilibria, and Oscillators

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

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Summary

A potential energy diagram shows how a system’s potential energy changes with position. By comparing the curve with the system’s constant mechanical energy, you can identify where motion is possible, where it reverses direction, and how kinetic and potential energy are exchanged.

For a system with conserved mechanical energy EE,

so the kinetic energy at position xx is K=E−U(x)K = E - U(x). Since kinetic energy cannot be negative, the particle can be only where U(x)≤EU(x) \leq E. On a diagram, these positions are where the potential energy curve lies at or below the horizontal line representing EE.

All flashcards

Potential energy diagram
A graph of potential energy as a function of position.
Allowed position
A position where the potential energy does not exceed total energy.
Turning point
A position where kinetic energy is zero and motion reverses direction.
Stable equilibrium
An equilibrium at a local potential minimum, where small displacements produce restoring forces.
Infinite potential well
A potential that rises without bound on both sides, confining finite-energy motion.

Quiz questions

  1. For one-dimensional conservative motion with potential energy U(x)U(x), what is the force?Answer: F(x)=−dUdxF(x)=-\frac{dU}{dx}
  2. At a smooth equilibrium position xex_e, which condition identifies a stable local minimum?Answer: U′(xe)=0U'(x_e)=0 and U′′(xe)>0U''(x_e)>0
  3. For U(x)=2(x4−x2)U(x)=2(x^4-x^2) J and total energy E=−0.25E=-0.25 J, why is x=0x=0 inaccessible?Answer: There, U(0)=0U(0)=0 exceeds the total energy.
  4. What is the frequency of an ideal spring oscillator with spring constant kk and mass mm?Answer: f=12πkmf=\frac{1}{2\pi}\sqrt{\frac{k}{m}}
  5. A spring oscillator starts from rest at its positive turning point. Which equation describes its position?Answer: x(t)=Acos⁡(ωt)x(t)=A\cos(\omega t)

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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