Flashcards — Chapter 8 🃏

Potential Energy and Conservation — the most powerful tools in physics! Master potential energy, conservative forces, energy conservation, and energy diagrams with these cards. ⚡

🎮 How it works

• 🎴 Study Mode — Cards one at a time. Think, then see the answer.
• ⚡ Quick Quiz — 10 questions in 20 seconds. Score, streak, and competition!

---

---

📋 Card deck: Key terms & formulas

Section §8.1 — Potential energy and conservative forces

1. Potential energy (U) — energy stored in a system due to the configuration of its parts; \( \Delta U = -W_{\text{conservative}} \) (joule, J)
2. Conservative force — a force whose work is independent of the path taken; potential energy can be defined for it
3. Relation between force and potential — \( \mathbf{F} = -\nabla U = -\frac{dU}{dx}\hat{x} \) (one dimension); force points toward decreasing U
4. Change in potential energy — \( \Delta U = U_f - U_i = -W_{\text{conservative}} \) (work done against the force)
5. Non-conservative force — a force whose work depends on the path (friction, air resistance)
6. Test for conservative force — if \( \nabla \times \mathbf{F} = 0 \) in 3D space, the force is conservative

Section §8.2 — Gravitational potential energy

7. Gravitational PE (near Earth) — \( U = mgh \) (relative to reference point \( h = 0 \)); valid for \( h \ll R_E \)
8. Gravitational PE (general) — \( U = -\frac{GMm}{r} \) (zero at infinity); exact at any distance
9. Linear approximation — for small \( h \), \( \Delta U \approx mgh \); comes from Taylor expansion of the exact formula
10. PE vs distance — for distant gravity, \( U(r) = -\frac{GMm}{r} \to 0 \) as \( r \to \infty \); always negative or zero

Section §8.3 — Elastic potential energy

11. Elastic PE (ideal spring) — \( U = \frac{1}{2}kx^2 \) (joule); \( x \) is displacement from equilibrium
12. Nonlinear spring — if \( F \neq -kx \), then \( U = \int F \, dx \) (integral of force)
13. Energy graph \( U(x) \) for spring — parabola; minimum at \( x = 0 \) (equilibrium)
14. Work by spring — \( W_{\text{spring}} = -\Delta U = \frac{1}{2}kx_i^2 - \frac{1}{2}kx_f^2 \)

Section §8.4 — Energy diagrams and motion analysis

15. Energy diagram — graphical display of \( U(x) \) and \( E_{\text{total}} \) (horizontal line) superimposed; vertical gap = \( E_{\text{total}} - U \) = kinetic energy
16. Equilibrium point — location where \( dU/dx = 0 \) (local min or max of U); force is zero there
17. Stable equilibrium — minimum of potential energy; small perturbation → particle returns
18. Unstable equilibrium — maximum of potential energy; small perturbation → particle flies away
19. Turning point — where \( E_{\text{total}} = U(x) \), so \( KE = 0 \) and \( v = 0 \); particle "reverses" here
20. Escape from potential well — if \( E_{\text{total}} > U_{\text{max}} \), particle can escape (leave the potential well)

Section §8.5 — Conservation of mechanical energy (extended)

21. Total mechanical energy — \( E_{\text{mech}} = KE + U = \frac{1}{2}mv^2 + U(x) \) (constant if no non-conservative forces)
22. Conservation of mechanical energy — \( E_i = E_f \) or \( KE_i + U_i = KE_f + U_f \); holds only for conservative forces
23. Differential form — \( \frac{dE}{dt} = \frac{d(KE + U)}{dt} = 0 \) if \( F = -dU/dx \)
24. Work by non-conservative force — \( W_{\text{non-cons}} = \Delta E_{\text{mech}} = E_f - E_i \); negative means energy dissipated
25. Energy loss to friction — \( \Delta E_{\text{mech}} = -f \cdot d \) (energy converted to heat and sound)

Section §8.6 — Multiparticle systems

26. Interaction potential energy — \( U_{12} = -\frac{Gm_1m_2}{r_{12}} \) (two masses); depends on separation
27. Total system energy — \( E = KE_1 + KE_2 + U_{12} \) (two-body system); conserved in isolated system
28. Center of mass and reference frames — \( KE_{\text{total}} = KE_{\text{CM}} + KE_{\text{rel}} \) (CM energy + relative motion energy)

Section §8.7 — Applications: orbits and escape

29. Orbital speed (circular orbit) — for circular orbit, \( \frac{Gm_1m_2}{r} = \frac{m_2v^2}{r} \), giving \( v = \sqrt{GM/r} \)
30. Orbital energy — \( E = -\frac{Gm_1m_2}{2r} \) (negative for bound orbits); must add energy to escape
31. Escape velocity — minimum speed to completely escape gravity; \( v_{\text{esc}} = \sqrt{2GM/r} \)
32. Derivation of escape velocity — if \( E_{\text{total}} = 0 \) at infinity, then now \( \frac{1}{2}mv^2 - \frac{GMm}{r} = 0 \), giving \( v = \sqrt{2GM/r} \)
33. Elliptical orbit — semi-major axis \( a \) determines the energy; circular orbit is special case with \( a = r \)
34. Perihelion and aphelion — closest and farthest points in orbit; speeds determined by energy conservation
35. Three-body problem — energy conserved but paths are unpredictable
36. Oscillations and energy — in simple harmonic motion, energy alternates between KE and PE; \( E = \frac{1}{2}kA^2 \) (A = amplitude)

Additional section — Important equations

37. Conservation law — \( \frac{d}{dt}(KE + U) = 0 \) if no non-conservative forces
38. Test for conservative (3D) — force is conservative ⟺ \( \nabla \times \mathbf{F} = \mathbf{0} \)
39. Equation of motion from potential — equations can be derived from potential: \( m\frac{d^2x}{dt^2} = -\frac{dU}{dx} \)
40. Conserved quantities — total energy, linear momentum (if no external force), angular momentum (if no external torque)

Additional section — Physical concepts

41. Isolated system — system with no external forces; energy and momentum are conserved
42. Force field — description of force at every point in space due to an extended source (gravity, electric field)
43. Potential (per unit mass) — \( V = U/m \) (potential energy per unit mass); property of field, not of object
44. Lagrangian — \( \mathcal{L} = KE - U \) (in calculus of variations); advanced tool for equations of motion
45. Energy-time relation — energy \( \times \) time = action; basis of quantum mechanics

---

📊 Topics covered

• Potential energy — definition, relation to force, path independence
• Conservative and non-conservative forces — tests, distinctions
• Gravitational PE — linear approximation and exact form
• Elastic PE — ideal and nonlinear springs
• Energy diagrams — equilibrium, stability, turning points, escape
• Energy conservation — proof, conditions, role of non-conservative forces
• Multiparticle systems — interaction energy
• Orbits and escape — orbital speed, escape velocity, elliptical orbits

---

What's next:

👉 §8.8 — Worked problems (step-by-step examples)
👉 §8.9 — Practice problems (exercises)
👉 §8.10 — Q&A (FAQs on common misconceptions)

⇧ Back to chapter