Short on time? This page is the recap. For explanations, graphs and worked examples, use the full study guide. This is the algebra-based course: no calculus is needed.
Recap
- Potential energy (U) is energy stored in the arrangement of a system whose objects interact through conservative forces (gravity, ideal springs).
- A single object cannot have potential energy. Name the pair: ball–Earth, block–spring.
- A conservative force does the same work on any path between two points. Friction is not conservative, so there is no friction potential energy.
- U is a scalar fixed by the positions of the objects in the system.
- The zero of U is your choice. Pick whatever makes the problem simplest and state it.
- For three or more objects, add the potential energy of every pair.
Key relationships
| Relationship | Symbols and units | Use it to |
|---|---|---|
| U_s = ½kΔx² | k in N/m, Δx in m from the relaxed length | find spring energy (never negative) |
| U_g = −Gm₁m₂ / r | r centre to centre in m; zero at infinity | find gravitational energy of two spheres (always negative) |
| ΔU_g = mgΔy | Δy in m, + for upward; g = 9.8 N/kg | find changes near a planet’s surface |
| ΔU = −W_conservative | J | link to work by a force inside the system |
| U_total = sum over pairs | J | three or more interacting objects |
Factors of change: Δx × 2 → U_s × 4. r × 2 → U_g × ½ (less negative, so larger). Δy × 2 → ΔU_g × 2.
Assumptions behind the numbers
- The spring is ideal: it obeys Hooke’s law and has negligible mass.
- Planets, moons and stars are treated as spherical, so r is measured between centres.
- ΔU_g = mgΔy assumes a constant field. It is accurate only when Δy is much smaller than the planet’s radius.
- g = 9.8 N/kg at Earth’s surface. The course framework says exam questions use 10 N/kg (10 m/s²), and 9.8 is also accepted.
Mistakes to avoid
- Giving potential energy to one object. Write “the ball–Earth system”.
- Forgetting to square Δx, or using total spring length instead of the change from relaxed length.
- Leaving Δx in centimetres. Convert to metres first.
- Measuring r from the surface in U_g = −Gm₁m₂/r.
- Calling U_g = −5 J “less” than U_g = −10 J. −5 J is the larger value.
- Using mgΔy for satellites. Use the general form when Δy is comparable with the radius.
- Missing a pair in a three-object system: A, B and C have three pairs.
Quick self-check
- A spring with k = 400 N/m is stretched 5.0 cm. How much energy does it store? (0.50 J)
- A 2.0 kg box is lifted 3.0 m. What is ΔU_g for the box–Earth system? (+59 J)
- Two asteroids move to three times their original separation. What happens to U_g? (It becomes one third of the original value: still negative, but larger.)
- Can the ball–Earth system have negative potential energy near the ground? (Yes, if the ball is below the chosen zero.)
Next: practice questions.