Work, Energy and Power · Lesson 9 of 16
Reading Potential-Energy Graphs
“Use the energy line and potential curve to identify allowed regions, turning points and equilibrium.”
• Read speed and force from an energy diagram. • Distinguish turning points, equilibria and barriers. • Apply the nonnegative kinetic-energy condition.
One graph gives two kinds of information
A potential-energy curve gives U as a function of position. Its slope gives the conservative force. If mechanical energy is conserved, draw a horizontal line at E. The vertical difference E − U(x) gives kinetic energy, so it determines speed. These are distinct readings: slope gives force, while vertical gap gives kinetic energy. The potential curve is an energy landscape, not the physical shape of a track.
A larger gap means faster motion; zero gap means zero instantaneous speed. At a boundary where E = U and the force drives the particle back into an allowed region, the particle turns. Equality alone does not prove a turn: a particle placed exactly at rest at a smooth equilibrium can remain there, and an ideal particle at a barrier top may approach it asymptotically. Energy tells which regions are allowed, but an initial condition selects which connected region the particle occupies.
A spring well and its turning points
Problem
A particle has U(x) = ½kx² with k = 0.5 N/m and total energy 1 J. Find the turning points.
- 1.K = 1 − 0.25x² in joules when x is in metres.
- 2.Physical motion requires 1 − 0.25x² ≥ 0, hence −2 m ≤ x ≤ 2 m.
- 3.At x = ±2 m, K = 0 and Fx = −kx is nonzero and directed back into the interval. These are turning points.
- 4.At x = 0, force is zero but speed is greatest for this energy. Zero force is not the same as zero speed.
Equilibrium and stability
At a minimum, moving a little to either side raises U, and the force points back toward the minimum. At a maximum, a small displacement lowers U and the force points farther away. A perfectly flat interval has zero force; a particle at rest there has neutral equilibrium in that ideal interval. The minimum possible classical total energy for a conservative potential is its lowest potential value, if that minimum exists; a particle at rest at a minimum realises it.
Steps, barriers and wells in the textbook diagrams
| Potential shape in the source | For the marked energy E | Lowest possible energy |
|---|---|---|
| Step: U = 0 on the left, U = V₀ on the right, with 0 < E < V₀ | x > a is forbidden | 0 |
| Positive staircase, with E below its lowest step | Every position is forbidden at that marked E | V₁, the lowest plateau |
| Square well U = −V₁ between a and b, with outer U = V₀ > E | x < a and x > b are forbidden | −V₁ |
| Central well U = −V₁, raised side regions and outer decreasing tails | The side bands between a/2 and b/2 and between −b/2 and −a/2 are forbidden where U > E | −V₁ |
In the last diagram, central and outer regions may both be allowed but separated by forbidden barriers. A particle initially in the central well cannot classically cross the barriers at that energy. At an ideal discontinuous step, interpret the potential on each side; the ordinary finite derivative formula does not apply exactly at the jump. Simple physical models include a gravity-height step, a sequence of height plateaus, a bound interaction well and a well separated from the outside by barriers. Quantum tunnelling is outside this classical treatment.
Check your understanding
Quiz
The gap E−U equals
Explanation: It is the energy left for motion.
A region with U>E is
Explanation: It would require negative kinetic energy.
A minimum of U usually corresponds to
Explanation: Small displacements produce restoring forces.
At a turning point in a spring well, acceleration is
Explanation: Velocity is zero but the restoring force is nonzero.
Two allowed regions separated by U>E are
Explanation: An energy barrier prevents passage without additional energy.
Key Takeaways
• Slope gives force; vertical energy gap gives speed. • Use U≤E to identify accessible regions. • Turning points and equilibrium points describe different conditions.