Potential energy diagrams show how the potential energy U of a system changes with position x. They are useful because they let you predict motion without solving the full force equation at every point. From one graph, you can identify where an object speeds up, slows down, stops, or stays in equilibrium.
These diagrams appear in mechanics, molecular physics, oscillations, and energy conservation problems.
The key connection is that force is the negative slope of the potential energy curve, F(x) = -dU/dx. Where the curve slopes downward to the right, the force points in the positive x direction, and where it slopes upward to the right, the force points in the negative x direction. Equilibrium occurs where the slope is zero, and stability depends on whether the point is a valley, a peak, or a flat region.
Turning points occur where the total mechanical energy line intersects U(x), because the kinetic energy is zero there.
Key Facts
- Total mechanical energy is E = K + U.
- Kinetic energy on a potential energy diagram is K = E - U(x).
- Motion is allowed only where E >= U(x).
- Force is the negative slope of potential energy: F(x) = -dU/dx.
- Equilibrium occurs where dU/dx = 0, so F = 0.
- Turning points occur where E = U(x), so K = 0 and the object reverses direction.
Vocabulary
- Potential energy diagram
- A graph of potential energy U versus position x that shows how energy and force depend on location.
- Total mechanical energy
- The sum of kinetic energy and potential energy, written E = K + U, for a system with no nonconservative work.
- Equilibrium point
- A position where the net force is zero because the slope of the potential energy curve is zero.
- Stable equilibrium
- An equilibrium point at a local minimum of U where a small displacement produces a restoring force back toward the point.
- Turning point
- A position where the object has zero kinetic energy and changes direction because E = U(x).
Common Mistakes to Avoid
- Confusing the height of U(x) with the force is wrong because force depends on the slope of the graph, not the value of potential energy.
- Using F = dU/dx is wrong because the correct relationship is F = -dU/dx, so the force points opposite the direction of increasing potential energy.
- Allowing motion where U(x) > E is wrong because K = E - U(x) would be negative, which is not physically possible for ordinary mechanical motion.
- Calling every equilibrium point stable is wrong because a local maximum is unstable and a flat equilibrium may require more information to classify.
Practice Questions
- 1 At a position x, a particle has total energy E = 12 J and potential energy U = 7 J. What is its kinetic energy at that position?
- 2 Near x = 2.0 m, the potential energy changes from U = 10 J at x = 2.0 m to U = 16 J at x = 4.0 m. Estimate the force over this interval using F = -ΔU/Δx.
- 3 A potential energy curve has a local minimum at x = 1 m and a local maximum at x = 5 m. Explain which point is stable, which is unstable, and how the force behaves after a small displacement from each point.