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.
Understanding Physics: Potential Energy Diagrams
The steepness of a curve tells you more than its direction. A steep section produces a larger force magnitude than a gentle section at the same position. This means an object has a greater acceleration there if its mass stays the same.
A horizontal section produces no force from that particular potential. Students often confuse height on the graph with the object’s physical height.
It is only an energy value. The horizontal axis is the chosen position coordinate, which might be distance along a track, the stretch of a spring, or the separation between two atoms.
A valley represents stable equilibrium because a small displacement creates a force back toward the bottom. An object placed near the bottom usually moves back and forth, exchanging kinetic energy with potential energy. For small motions near a smooth, rounded valley, the motion can be close to simple harmonic motion.
A narrower valley has more sharply changing slopes, so the restoring force grows quickly with displacement. It tends to give faster oscillations. A broad valley gives weaker restoring forces near its bottom.
A peak is unstable equilibrium. A tiny displacement from its exact top leads to a force that pushes the object farther away.
The same curve can describe very different motions depending on the total energy. A low energy object may be trapped inside one valley because surrounding hills are too high to cross. Raising its energy lets it pass over a barrier and enter another region.
If the curve rises without limit on both sides, the object remains confined for any finite energy. If the curve falls toward a constant value far away, energy above that value can allow escape to large distances. This idea appears in atomic and molecular physics.
Atoms in a molecule have a preferred separation at the bottom of an energy well. Supplying enough energy can separate them by breaking the bond.
Potential energy diagrams work most cleanly when friction and air resistance are negligible. With these effects, mechanical energy decreases as motion proceeds, so one fixed horizontal energy level no longer describes the whole journey. The object can eventually settle at the bottom of a valley.
The zero level of potential energy is a choice, not a physical location with special meaning. Shifting every potential energy value upward or downward changes no force and no motion, as long as total energy is shifted by the same amount. When reading a graph, first identify the coordinate, then inspect slopes, valleys, peaks, barriers, and the stated energy level.
Keep track of whether the graph describes one object or a whole interacting system. This prevents many common mistakes in spring, gravity, and particle motion problems.
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.