Potential energy Study Guide

📖 Core Concepts Potential Energy (PE) – Energy stored due to an object’s position or configuration (e.g., height, spring stretch, charge separation). Measured in joules (J). Conservative Force – A force whose work depends only on start and end points, not on the path taken (e.g., gravity, spring force, electrostatic force). Scalar Potential – A scalar field \(U(\mathbf{r})\) whose gradient gives the force: \(\mathbf{F}= -\nabla U\). Energy Conservation in a Bow‑and‑Arrow – Chemical energy in the archer’s muscles → elastic PE in the bow → kinetic energy of the arrow; total energy stays constant. Reference Level – PE is defined relative to an arbitrarily chosen zero; keep the same reference throughout a problem. 📌 Must Remember Work–Energy Relation: \(W = \displaystyle\int{C}\mathbf{F}\!\cdot d\mathbf{r}=U(A)-U(B)\). Force–PE Sign Convention: Positive work by the force lowers PE; work against the force raises PE. Key Formulas Spring: \(U = \tfrac{1}{2}k x^{2}\) Gravitational (near Earth): \(U = m g h\) Gravitational (two bodies): \(U = -\dfrac{G M m}{r}\) Electrostatic: \(U = \dfrac{1}{4\pi\varepsilon{0}}\dfrac{Q q}{r}\) Capacitor: \(U = \tfrac{1}{2} C V^{2}\) Magnetic dipole: \(U = -\mathbf{m}\!\cdot\!\mathbf{B}\) Units: 1 J = 1 N·m = 1 kg·m²/s². 🔄 Key Processes Identify the conservative force involved (gravity, spring, electric, magnetic). Write the force expression (e.g., \(\mathbf{F}= -k x\,\hat{x}\) for a spring). Integrate the force or use a known formula to obtain PE: \[ U = -\int \mathbf{F}\cdot d\mathbf{r} \] Choose a reference point (often \(U=0\) at \(x=0\), \(h=0\), or infinite separation). Apply energy conservation: \[ \text{Initial chemical/PE} = \text{Final KE} + \text{any remaining PE} \] Check sign consistency with the chosen reference. 🔍 Key Comparisons Spring PE vs. Gravitational PE (near Earth) Spring: \(U = \tfrac{1}{2}k x^{2}\) (quadratic in displacement) Gravity: \(U = m g h\) (linear in height) Attractive vs. Repulsive Conservative Forces Attractive (gravity, electrostatic opposite charges): \(U\) is negative and becomes less negative as objects separate. Repulsive (like charges, spring compression): \(U\) is positive and grows with separation/compression. Reference Zero at Surface vs. Infinity Surface: \(U=0\) at \(h=0\) → \(U=mgh\) positive for heights above ground. Infinity: \(U=0\) at \(r\to\infty\) → gravitational two‑body PE \(U=-GMm/r\) negative near Earth. ⚠️ Common Misunderstandings “PE is always positive.” PE can be negative (gravitational two‑body, electrostatic opposite charges) depending on the reference. Confusing work done by a force with work done against it. Remember the sign convention: work by a conservative force lowers PE. Treating the bow’s elastic PE as \(\tfrac{1}{2}k x^{2}\) without checking linearity. Real bows are not perfectly linear springs; the formula is an approximation. 🧠 Mental Models / Intuition Energy Landscape: Visualize PE as a hill‑shaped surface. Objects roll downhill (force does positive work) and climb uphill (work must be supplied). “Zero at Infinity” Trick: For any attractive inverse‑square force, imagine pulling the objects infinitely far apart – the system has no stored PE there (zero). 🚩 Exceptions & Edge Cases Non‑linear bows or springs: \(U\neq \tfrac{1}{2}k x^{2}\) if the force–displacement relation deviates from Hooke’s law. Strong‑field gravity (near black holes): Newtonian \(U=-GMm/r\) no longer accurate; relativistic corrections apply. Time‑varying fields: If the electric or magnetic field changes with time, the simple \(U=-\mathbf{m}\!\cdot\!\mathbf{B}\) or electrostatic formulas need additional terms. 📍 When to Use Which Use \(U=\tfrac{1}{2}k x^{2}\) when the problem states a linear spring or Hooke’s‑law behavior. Use \(U=mgh\) for small height changes near Earth where \(g\) is essentially constant. Use \(U=-\dfrac{GMm}{r}\) for planetary‑scale or satellite problems where separation varies significantly. Use the capacitor formula \(\tfrac{1}{2}CV^{2}\) when dealing with stored electric energy in isolated capacitors, not in inductors. Use \(U=-\mathbf{m}\!\cdot\!\mathbf{B}\) for magnetic dipoles in static magnetic fields (e.g., compass needle). 👀 Patterns to Recognize Quadratic dependence → spring or capacitor. Look for a squared term → likely \(\tfrac{1}{2}k x^{2}\) or \(\tfrac{1}{2}CV^{2}\). Linear dependence on height → near‑Earth gravity. If only \(h\) appears, think \(U=mgh\). Inverse‑distance dependence → gravitational or electrostatic interaction. Presence of \(1/r\) signals \(U\propto -1/r\) (attractive) or \(+1/r\) (repulsive). Dot product with field → magnetic dipole. If you see \(\mathbf{m}\cdot\mathbf{B}\), that’s magnetic PE. 🗂️ Exam Traps Choosing the wrong reference zero → leads to sign errors (e.g., reporting gravitational PE as positive when the textbook uses zero at infinity). Mixing up work done by vs. against the force → a common distractor flips the sign of \(\Delta U\). Applying \(\tfrac{1}{2}k x^{2}\) to a non‑linear bow → answer will be off; the problem may give a specific force‑displacement curve. Assuming PE is always stored energy – some “potential” terms (e.g., magnetic dipole) can be negative; the magnitude, not the sign, often matters for energy released. Forgetting the factor ½ in capacitor and spring formulas – many multiple‑choice options omit it to test attention to detail. --- Review each bullet before the exam; the formulas and sign conventions are the quickest way to earn points on any potential‑energy problem.