Potential Energy Curves & Stability

Introduction

In classical mechanics and modern physics, the concepts of potential energy curves and stability play a central role. Whether we are analyzing how a pendulum swings, how atoms bond in molecules, or how planets orbit the Sun, potential energy diagrams provide an elegant way to visualize and predict motion.

A potential energy curve is simply a graph that shows how potential energy varies with position or configuration. By studying the shape of this curve, we can determine whether a system is in stable equilibrium, unstable equilibrium, or neutral equilibrium.

This article explores:

• The meaning of potential energy curves.
• The mathematics behind equilibrium and stability.
• How forces relate to the slope of potential energy curves.
• Examples from mechanics, chemistry, and astrophysics.
• Real-world applications, solved problems, and misconceptions.

By the end, you’ll see how a simple curve can reveal the “future” behavior of physical systems.

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What is a Potential Energy Curve?

A potential energy curve is a graph where:

• The x-axis represents position (or configuration).
• The y-axis represents potential energy (U).

By examining the shape of this curve, we can determine:

• Where equilibrium positions lie.
• Whether these equilibria are stable or unstable.
• How particles will move when displaced from equilibrium.

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Connection Between Force and Potential Energy

Force is directly related to the slope of the potential energy curve: F(x)=−dUdxF(x) = – \frac{dU}{dx}F(x)=−dxdU​

• If slope dUdx>0\frac{dU}{dx} > 0dxdU​>0: force is negative (pulls left).
• If slope dUdx<0\frac{dU}{dx} < 0dxdU​<0: force is positive (pushes right).
• At equilibrium: dUdx=0\frac{dU}{dx} = 0dxdU​=0.

So, equilibrium points occur at minima, maxima, or flat regions of the potential energy curve.

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Equilibrium and Stability

1. Stable Equilibrium

Occurs at a minimum point of the potential energy curve.

Conditions:

• dUdx=0\frac{dU}{dx} = 0dxdU​=0
• d2Udx2>0\frac{d^2U}{dx^2} > 0dx2d2U​>0 (curve concave upward).

Interpretation:
If displaced slightly, the system tends to return to equilibrium.

Examples:

• A ball at the bottom of a bowl.
• Atoms bonded at equilibrium bond length.

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2. Unstable Equilibrium

Occurs at a maximum point of the potential energy curve.

Conditions:

• dUdx=0\frac{dU}{dx} = 0dxdU​=0
• d2Udx2<0\frac{d^2U}{dx^2} < 0dx2d2U​<0 (curve concave downward).

Interpretation:
If displaced slightly, the system moves further away from equilibrium.

Examples:

• A ball balanced at the top of a hill.
• Inverted pendulum at topmost position.

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3. Neutral Equilibrium

Occurs on a flat region of the potential energy curve.

Conditions:

• dUdx=0\frac{dU}{dx} = 0dxdU​=0
• d2Udx2=0\frac{d^2U}{dx^2} = 0dx2d2U​=0.

Interpretation:
If displaced, the system stays at the new position (neither returns nor moves away).

Examples:

• A ball on a flat horizontal surface.
• A satellite in deep space far from gravitational fields.

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Graphical Examples

1. Parabolic Curve (U = ½kx²):
   • Represents a spring or harmonic oscillator.
   • Minimum at x = 0 → stable equilibrium.
2. Inverted Parabola (U = -½kx²):
   • Represents unstable equilibrium at x = 0.
3. Double Well Potential:
   • Two stable equilibria separated by an unstable one.
   • Used in chemistry (bonding), quantum mechanics (tunneling).
4. Gravitational Potential Curve:
   • U=−GMm/rU = -GMm/rU=−GMm/r, shows attractive nature of gravity.
   • Stable orbits correspond to minima of effective potential.

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Potential Energy Curves in Mechanics

1. Simple Pendulum

• Potential energy: U=mgh=mgL(1−cos⁡θ)U = mgh = mgL(1 – \cos \theta)U=mgh=mgL(1−cosθ).
• Minimum at θ=0\theta = 0θ=0 (stable equilibrium).
• Maximum at θ=π\theta = \piθ=π (unstable equilibrium).

2. Mass-Spring System

• Potential energy: U=½kx2U = ½kx²U=½kx2.
• Curve is parabolic with a stable equilibrium at x = 0.

3. Gravitational Systems

• Near Earth: U=mghU = mghU=mgh.
• For planets: U=−GMm/rU = -GMm/rU=−GMm/r.
• Effective potential (with angular momentum) gives stable orbits.

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Potential Energy Curves in Chemistry

• Atoms interact via interatomic potential curves.
• Morse Potential: U(r)=De(1−e−a(r−re))2U(r) = D_e \left(1 – e^{-a(r-r_e)}\right)^2U(r)=De​(1−e−a(r−re​))2
  • rer_ere​: equilibrium bond length (stable).
  • DeD_eDe​: bond energy (depth of potential well).
• Explains bonding, vibrations, and molecular stability.

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Stability in Terms of Energy

• Stable equilibrium: energy minimum → system resists displacement.
• Unstable equilibrium: energy maximum → slightest push breaks equilibrium.
• Neutral equilibrium: flat → system indifferent to displacement.

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Real-World Applications

1. Engineering: Designing stable structures (bridges, towers) requires potential energy analysis.
2. Robotics: Balance and motion stability are modeled with energy curves.
3. Chemistry: Chemical bonding and reactions depend on potential energy wells.
4. Astrophysics: Orbital stability of planets and satellites.
5. Civil Engineering: Stability of slopes and dams modeled with energy methods.
6. Quantum Mechanics: Potential wells explain tunneling and electron energy levels.
7. Biology: Protein folding governed by potential energy landscapes.

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Misconceptions

1. All equilibrium positions are stable.
   • False. Many are unstable (e.g., top of hill).
2. Potential energy must always be positive.
   • Wrong. Gravitational potential energy can be negative.
3. If slope of curve = 0, system is safe.
   • Not necessarily → must check second derivative.
4. Path affects stability.
   • Stability depends only on energy landscape, not path.

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Historical Notes

• Newton introduced potential concepts via gravity.
• Joseph Louis Lagrange and William Rankine formalized energy curves.
• In 20th century, potential energy surfaces became vital in chemistry and quantum physics.

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Solved Problems

Problem 1: Spring System

A 2 kg mass attached to a spring with k = 200 N/m is displaced by 0.1 m.
Find potential energy and stability. U=½kx2=½(200)(0.1)2=1 JU = ½kx² = ½(200)(0.1)² = 1 \, JU=½kx2=½(200)(0.1)2=1J

Stable equilibrium at x = 0.

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Problem 2: Pendulum Stability

At what angle does a pendulum become unstable?

At θ = 180° (inverted position). Curve has maximum → unstable equilibrium.

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Problem 3: Planet Orbit

For Earth-Sun system, find condition for stable circular orbit.

Condition: derivative of effective potential = 0.
Gives balance between centrifugal and gravitational forces.

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Problem 4: Double Well Potential

Explain stability of molecule with two equilibrium positions.

Answer: Two minima → two stable states. Particle can shift if it gains energy above barrier.

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Practice Questions

1. Explain why a ball inside a bowl has stable equilibrium.
2. Draw and explain potential energy curve for a pendulum.
3. Why is gravitational potential energy negative?
4. Differentiate between stable and neutral equilibrium with examples.
5. Describe how chemical bonds relate to potential energy wells.