Short on time? This page is the recap for the calculus-based course (separate from Physics 1). For derivations, the U(x) graph and worked examples, use the full study guide.
Recap
- Potential energy is a scalar stored in the arrangement of a system of two or more objects.
- It exists only for conservative interactions: work independent of path, zero around a closed loop. Friction has no potential energy.
- You choose where U = 0. Only changes in U are physical.
- The conservative force points towards lower U.
- For three or more objects, add the energy of every pair once.
Key relationships
| Relationship | Units | Notes |
|---|---|---|
| ΔU = −W_c = −∫F_c · dr | J | defines U from a conservative force |
| F_x = −dU/dx | N | minus the slope of the U(x) graph |
| dU/dx = 0 | — | equilibrium position |
| d²U/dx² > 0 / < 0 | — | stable (minimum) / unstable (maximum) |
| U_s = ½k(Δx)² | J | ideal spring, U = 0 at relaxed length |
| U_g = −Gm₁m₂/r | J | spheres, U = 0 at infinite separation |
| ΔU_g ≈ mgΔy | J | near a surface, Δy much smaller than the radius |
| U = U₁₂ + U₁₃ + U₂₃ + … | J | systems of three or more objects |
Assumptions behind the numbers
- One-dimensional motion along a stated axis unless told otherwise.
- Ideal spring: massless, force proportional to deformation.
- Spheres act as point masses at their centres. g = 9.8 m/s²; G = 6.67 × 10⁻¹¹ N·m²/kg².
Mistakes to avoid
- “An object has potential energy.” A system does.
- Losing the minus sign in ΔU = −W_c or F_x = −dU/dx.
- Equilibrium where U = 0. Equilibrium is where the slope is zero.
- Stable at a maximum. A maximum pushes the object away.
- mgΔy for large heights. Use −Gm₁m₂/r.
- ½k(Δx)² for a nonideal spring. Integrate the actual force.
- Counting a pair twice in a many-body system.
Quick self-check
- U(x) = (4.0 J/m²)x². What is F_x at x = 0.50 m? (−dU/dx = −8.0x = −4.0 N)
- A spring has k = 200 N/m. How much does U_s rise when the stretch goes from 0.050 m to 0.10 m? (½ × 200 × (0.010 − 0.0025) = 0.75 J)
- U(x) has a local maximum at x = 3.0 m. Is that equilibrium stable? (No. A small push gives a force away from 3.0 m: unstable.)
Next: practice questions.