Gravitational PE: U = mgh

• h: height above the chosen reference level
• Reference level is arbitrary (choose for convenience)
• Can be positive, negative, or zero depending on reference
• Change in PE is independent of reference: $delta_U$ = mg*$delta_h$

Elastic (Spring) PE: U = \frac{1}{2}$$kx^2

• x: displacement from natural (relaxed) length
• Always >= 0 (for both compression and extension)
• Maximum PE at maximum compression/extension
• PE = 0 at natural length

Relation to Force: F = -dU/dx (force is negative gradient of PE)

• This means force points toward lower PE
• At equilibrium points: F = 0, dU/dx = 0
• Stable equilibrium: $d^{2U}$/$dx^2$ > 0 (PE minimum)
• Unstable equilibrium: $d^{2U}$/$dx^2$ < 0 (PE maximum)

PE diagrams (U vs x): Used to analyze motion qualitatively. The body oscillates between turning points where KE = 0 (total energy = PE).