Potential Energy & Force
A short lesson with diagrams, a worked example, a problem to try, and a quick check. Then quiz yourself on 6 practice questions.
For a conservative force, potential energy and force are two views of the same thing: F = −dU/dx and ΔU = −∫F dx. On a U(x) graph, the force points downhill, and equilibrium is wherever the slope is zero.
1. Reading a U(x) graph
2. Conservative forces
Gravity and springs are conservative: the work they do depends only on the start and end points, and is zero around any closed path. That’s what lets you define a U.
Friction isn’t: it does negative work on every part of a path, so a closed loop still costs energy.
3. Example, then try it
A particle’s potential energy is U(x) = x³ − 3x (J). Find the force at x = 2 m and the equilibrium points.
F = −dU/dx = −(3x² − 3) = 3 − 3x². At x = 2: F = 3 − 12 = −9 N (toward −x).
F = 0 at x = ±1 m: x = 1 is a minimum (stable), x = −1 a maximum (unstable).
A 0.50 kg particle moves in that potential with total energy E = 1 J. Find its speed at the minimum (x = 1 m) and at x = 0, and the force on it at x = 0. Where are its turning points?
At x = 1: U = −2 J, so K = 1 − (−2) = 3 J and v = √(2 × 3 ÷ 0.50) = √12 ≈ 3.5 m/s.
At x = 0: U = 0, so K = 1 J and v = √(2 × 1 ÷ 0.50) = 2.0 m/s.
Force at 0: F = 3 − 0 = +3 N, pointing downhill toward the minimum.
Turning points: where U = E = 1 J. From the graph, about x ≈ −0.35 m and x ≈ 1.9 m.
Answer: 3.5 m/s at the minimum, 2.0 m/s at x = 0, F = +3 N, turning near −0.35 m and 1.9 m.
4. Common mistakes
At a certain point, a particle’s U(x) graph slopes upward to the right. Which way does the force on it point?
F = −dU/dx. The slope is positive, so F is negative: the force pushes the particle downhill, toward −x.
Ready to practice?
6 questions on Potential Energy & Force, each with a hint and an explanation for every wrong answer.