7.2 Energy Diagrams and Equilibrium

Energy Diagrams

Energy diagrams plot a system's potential energy as a function of position. They let you visually determine where an object will speed up, slow down, turn around, or sit in equilibrium, all without solving equations of motion. This section covers how to read these diagrams, classify equilibrium points, and use energy conservation to analyze motion.

Interpretation of Energy Diagrams

An energy diagram has position on the horizontal axis and potential energy $U(x)$ on the vertical axis. The shape of the curve tells you almost everything about how an object behaves at each location.

• Valleys (low points) are stable positions where objects naturally settle. Think of a marble rolling to the bottom of a bowl.
• Peaks (high points) are unstable positions. An object perched at a peak will move away from it with the slightest nudge.
• Flat regions (plateaus) represent neutral stability. An object placed anywhere on a plateau has no tendency to move one way or the other.
• Energy wells are valley-shaped regions of lower potential energy surrounded by higher energy on both sides. Objects can become trapped inside a well if they don't have enough total energy to climb out.
• Energy barriers are hill-shaped regions of higher potential energy sitting between two wells. An object needs enough total energy to get over a barrier before it can move from one well to another.

Classification of Equilibrium Points

An equilibrium point is any position where the net force on the object is zero. On the energy diagram, this corresponds to a spot where the slope of the curve is zero (a horizontal tangent line), because force relates to potential energy by $F = -\frac{dU}{dx}$.

• Stable equilibrium occurs at a local minimum (bottom of a valley). If you push the object slightly away, the force pushes it back. A ball resting at the bottom of a bowl is the classic example.
• Unstable equilibrium occurs at a local maximum (top of a hill). Any small disturbance causes the object to accelerate away from that point. Picture a ball balanced on top of an overturned bowl.
• Neutral equilibrium occurs on a flat region where $U(x)$ is constant. Displacing the object produces no restoring force and no repelling force; it just stays in its new position. A ball on a level table behaves this way.

Complex potential energy curves can have multiple equilibrium points. A double-well potential, for instance, has two stable minima separated by one unstable maximum.

Motion Analysis with Energy Diagrams

The key principle here is conservation of mechanical energy: in an isolated system (no friction or other non-conservative forces), the total energy stays constant.

$E_{total} = KE + U = \text{constant}$

You can draw a horizontal line on the energy diagram at the object's total energy $E_{total}$. The vertical gap between that line and the $U(x)$ curve at any position equals the kinetic energy at that position.

To analyze an object's motion on an energy diagram:

1. Draw a horizontal line at the system's total energy $E_{total}$.

2. At any position $x$, the kinetic energy is $KE = E_{total} - U(x)$. Since $KE$ can't be negative, the object can only exist where $E_{total} \geq U(x)$.

3. Turning points are positions where $E_{total} = U(x)$, so $KE = 0$. The object momentarily stops and reverses direction here.

4. The object moves fastest where $U(x)$ is lowest, because that's where $KE$ is greatest.

5. If the object is inside a well and $E_{total}$ is below the surrounding barriers, it oscillates back and forth between two turning points. Steeper wells produce higher-frequency oscillations.

6. If $E_{total}$ exceeds the height of a barrier, the object can pass over it and is not confined to one well.

Energy Types in Diagrams

• Potential energy $U(x)$ is read directly off the vertical axis. The choice of where $U = 0$ is arbitrary, but you must keep it consistent throughout a problem.
• Kinetic energy is the difference: $KE = E_{total} - U(x)$.
• Speed at any position follows from the kinetic energy: $v = \sqrt{\frac{2\,KE}{m}}$. The direction of motion (left or right) isn't given by the speed formula; you determine it from context or from the sign of the slope of $U(x)$, since force points in the direction that decreases $U$.
• At turning points, all energy is potential ($KE = 0$). At stable equilibrium, potential energy is at a minimum and kinetic energy is at its maximum for that well.
• At the total energy line, watch for the distinction: $E_{total}$ is constant for a closed system, but if non-conservative forces act, mechanical energy is not conserved and $E_{total}$ changes over time.