§8.6 — Energy Diagrams

An energy diagram is a graph of potential energy \( U(x) \) as a function of position. It encodes rich information about motion without requiring explicit solution of equations. Horizontal lines represent the total energy \( E \); the vertical distance from \( U \) to \( E \) gives the kinetic energy at that point. This section develops energy diagrams as a tool for predicting motion qualitatively, identifying turning points, and classifying equilibria as stable or unstable.

Reading an energy diagram

On an energy diagram:

• The vertical axis is energy (measured in joules)
• The horizontal axis is position (measured in meters or some other coordinate)
• A curve labeled \( U(x) \) is the potential energy
• A horizontal line labeled \( E \) is the total mechanical energy

At any position \( x \), the kinetic energy is:

\[ KE(x) = E - U(x) \]

Key observations:

1. If \( E > U(x) \), then \( KE > 0 \) — the object can reach position \( x \)
2. If \( E < U(x) \), then \( KE < 0 \) — impossible; the object cannot reach position \( x \) (classically forbidden)
3. If \( E = U(x) \), then \( KE = 0 \) — the object momentarily stops (turning point)

Turning points and allowed regions

Turning points occur where \( E = U(x) \). At these positions, all energy is potential; the object has zero velocity and reverses direction.

Allowed region: The range of positions where \( E \geq U(x) \). The object can only move within this region.

Forbidden region: Where \( E < U(x) \). Classically, the object cannot penetrate these regions (though quantum mechanically, tunneling is possible).

Example: Spring potential (\( U = \frac{1}{2}kx^2 \))

For a spring with \( k = 100 \) N/m and total energy \( E = 5 \) J:

At the turning points: \[ 5 = \frac{1}{2} \times 100 \times x^2 \implies x = \pm 0.316 \text{ m} \]

The object oscillates between \( x = -0.316 \) m and \( x = +0.316 \) m. The amplitude is \( A = 0.316 \) m.

At equilibrium (\( x = 0 \)): \( U(0) = 0 \), so \( KE = E - 0 = 5 \) J. The object has maximum kinetic energy (maximum speed) here.

At the turning points (\( x = \pm 0.316 \) m): \( U = 5 \) J, so \( KE = 0 \). The velocity is zero; the object reverses direction.

Equilibrium and stability

An equilibrium point is where the force is zero: \( F = -dU/dx = 0 \). This occurs at extrema of \( U \):

• Minimum of \( U \): Stable equilibrium
• Maximum of \( U \): Unstable equilibrium
• Inflection point: Neutral equilibrium (rare)

Graphical method for stability

On an energy diagram:

1. Find equilibrium points (where \( dU/dx = 0 \), i.e., where the curve has horizontal tangent)
2. At a minimum, the curve is concave up: a small perturbation creates a restoring force (stable)
3. At a maximum, the curve is concave down: a small perturbation creates a force pushing away (unstable)

Example: For \( U(x) = x^4 - 4x^2 \) (two potential wells):

• Minima at \( x = \pm \sqrt{2} \): stable equilibria
• Maximum at \( x = 0 \): unstable equilibrium

An object released near a minimum oscillates around it; released near the maximum, it rolls away.

Bounded and unbounded motion

Bound motion: If \( E \) is below the height of all "walls" in the potential, the object is trapped and oscillates back and forth. The amplitude is determined by where \( E = U \).

Unbound motion: If \( E \) is above the potential barrier, the object has enough energy to escape to infinity. It continues moving without returning.

Example: Particle in a potential well

Consider \( U(x) = -U_0 \exp(-x^2/a^2) \) (an attractive potential well).

• If \( E < 0 \): The object is bound, oscillating within the well
• If \( E \geq 0 \): The object can escape to infinity; motion is unbound

The critical energy \( E = 0 \) separates bound from unbound regimes.

Speed from the energy diagram

The speed at position \( x \) can be read from the diagram:

\[ KE = E - U(x) = \frac{1}{2}mv^2 \]

\[ v(x) = \sqrt{\frac{2(E - U(x))}{m}} \]

Large gap between \( E \) and \( U(x) \) → large kinetic energy → high speed. Small gap → low speed. Zero gap (\( E = U(x) \)) → zero speed (turning point).

Example: Block on a curved surface

A block slides on a smooth curve with shape \( U(x) = x^2 \) (paraboloid). If the block starts at rest at \( x_0 = 1 \) m with mass \( m = 1 \) kg:

Initial energy: \( E = U(1) + 0 = 1 \) J

Speed at \( x = 0 \) (bottom): \[ v(0) = \sqrt{\frac{2(E - U(0))}{m}} = \sqrt{\frac{2(1 - 0)}{1}} = \sqrt{2} \approx 1.41 \text{ m/s} \]

Speed at \( x = 0.5 \) m: \[ v(0.5) = \sqrt{\frac{2(1 - 0.25)}{1}} = \sqrt{1.5} \approx 1.22 \text{ m/s} \]

The speed increases as the block descends (lower potential energy), then decreases as it climbs the other side.

Oscillation frequency from the energy diagram

For a symmetric potential well near equilibrium, we can estimate the oscillation frequency from the curvature of \( U(x) \) near the minimum.

If \( U(x) \approx U_0 + \frac{1}{2}k(x - x_0)^2 \) near equilibrium \( x_0 \) (harmonic approximation), then:

\[ \omega = \sqrt{\frac{k}{m}} \]

The stiffer the potential (sharper curvature), the faster the oscillation.

Qualitative vs. quantitative analysis

Qualitative analysis from energy diagrams:

• Identify equilibrium points and their stability
• Find turning points and allowed regions
• Estimate oscillation frequency (curvature)
• Determine if motion is bound or unbound
• Compare speeds at different positions

Quantitative analysis requires:

• Solving \( F = ma \) (Newton's laws)
• Solving \( v(t) \) and \( x(t) \) explicitly
• Detailed integration of equations of motion

Energy diagrams provide the big picture without the algebra.

Connection to Chapter 7 (Work and Energy)

The energy diagram is the graphical manifestation of the work–energy theorem. The area under the force curve in an \( F \)-\( x \) plot equals the work; equivalently, it equals the change in potential energy visible on the energy diagram.

What you should be able to do

After this section, you should be able to:

• Sketch or interpret an energy diagram for a given potential \( U(x) \)
• Identify turning points, allowed regions, and forbidden regions from an energy diagram
• Determine equilibrium points and classify them as stable, unstable, or neutral
• Estimate the speed at any position using \( v = \sqrt{2(E - U)/m} \)
• Distinguish between bound and unbound motion
• Use curvature of the potential to estimate oscillation frequency
• Qualitatively predict the motion of a particle without solving equations

Preview of §8.7: Complex Systems

Real systems often involve multiple particles, multiple degrees of freedom, or constraints (like rigid bodies rotating). §8.7 extends energy methods to these complex scenarios, introducing concepts like reduced mass, center-of-mass motion, and constraint forces.

📚 See also: Halliday Vol 1, Ch 8, §8.7 — Potential Energy Diagrams. 📖 Open reference: OpenStax University Physics Vol 1 — Chapter 8.7: Energy Diagrams and Stability. 🎓 Video: MIT 8.01 Lecture 16 (Walter Lewin) — "Potential Energy and Turning Points".

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