Gravity Study Guide

📖 Core Concepts Gravity – universal attraction between any two masses; strength ∝ product of masses, ∝ 1 / \(r^{2}\). Newton’s Law of Universal Gravitation – \(F = G\frac{m{1}m{2}}{r^{2}}\) ( \(G = 6.674\times10^{-11}\ \mathrm{m^{3}kg^{-1}s^{-2}}\) ). Equivalence Principle – inertial mass = gravitational mass to better than 1 part × 10⁻¹²; free‑fall is locally indistinguishable from weightlessness. General Relativity (GR) – gravity = curvature of spacetime described by the Einstein field equation $$G{\mu\nu} + \Lambda g{\mu\nu}= \frac{8\pi G}{c^{4}}\,T{\mu\nu}.$$ Standard Gravity on Earth – average \(g = 9.80665\ \mathrm{m\,s^{-2}}\); varies 9.780 → 9.832 m s⁻² from equator to pole because of rotation. Orbital Motion – a body continuously “falls” toward the central mass while keeping enough tangential speed to stay in a closed (usually elliptical) path. Dark Matter & Dark Energy – unseen mass/energy inferred from galaxy rotation curves and accelerating cosmic expansion; both act only through gravity. --- 📌 Must Remember Inverse‑square law: \(F \propto 1/r^{2}\); halving distance ⇒ force × 4. Gravitational constant: \(G = 6.674\times10^{-11}\ \mathrm{m^{3}kg^{-1}s^{-2}}\). Escape velocity: \(v{\text{esc}} = \sqrt{2GM/R}\). Standard gravitational parameter: \(\mu = GM\) (used in orbital calculations). Key GR tests: light‑deflection (Eddington 1919), gravitational redshift (Pound–Rebka 1959), Shapiro time delay (1964), frame‑dragging (Gravity Probe B 2011), gravitational waves (LIGO 2015). Galaxy rotation problem: outer stars move faster than visible mass predicts → dark matter or Modified Newtonian Dynamics (MOND). Equivalence of inertial & gravitational mass → free‑fall acceleration is independent of object’s composition. --- 🔄 Key Processes Calculating Gravitational Force Identify masses \(m{1}, m{2}\) and separation \(r\). Plug into \(F = G m{1}m{2}/r^{2}\). Deriving Orbital Speed for Circular Orbit Set centripetal force \(mv^{2}/r\) equal to gravitational force \(GMm/r^{2}\). Solve: \(v = \sqrt{GM/r}\). Finding Escape Velocity Equate kinetic energy \(\frac{1}{2}mv{\text{esc}}^{2}\) to gravitational potential energy \(GMm/R\). Result: \(v{\text{esc}} = \sqrt{2GM/R}\). Applying Gauss’s Law for Gravity Compute flux \(\oint \mathbf{g}\cdot d\mathbf{A} = -4\pi G M{\text{enc}}\). For a spherically symmetric body, \(\mathbf{g}= -GM/r^{2}\,\hat{r}\) outside the mass. Transition from Newtonian to Relativistic Regime Use Newtonian formulas for \(v \ll c\) and weak fields. Switch to GR when dealing with strong fields (near black holes, high‑precision GPS, perihelion precession). --- 🔍 Key Comparisons Newtonian Gravity vs. General Relativity Force vs. Curvature: Newton – instantaneous force; GR – spacetime curvature, finite propagation speed \(c\). Range of Validity: Newton works for everyday speeds & weak fields; GR required for strong fields, high precision, or light propagation. Gravitational Mass vs. Inertial Mass Definition: Gravitational mass determines the strength of gravitational attraction; inertial mass resists acceleration. Reality: Experimentally identical to 1 part × 10⁻¹²; the equivalence principle treats them as the same. Dark Matter vs. Modified Newtonian Dynamics (MOND) Dark Matter: Adds unseen mass that interacts only gravitationally. MOND: Alters the force law at low accelerations (\(a < a{0}\)) to explain flat rotation curves without extra mass. --- ⚠️ Common Misunderstandings “Gravity is a force in GR.” In GR there is no force; objects follow geodesics in curved spacetime. “All objects fall at the same speed because of air resistance.” In vacuum, all objects accelerate at the same rate due to the equivalence principle; air resistance is the only cause of differences. “Black holes suck everything in.” The event horizon is a surface where escape speed exceeds \(c\); objects far enough away feel the same inverse‑square pull as any massive body. “Gravitational waves are like sound waves.” They are ripples in spacetime itself, not oscillations of a medium. --- 🧠 Mental Models / Intuition Rubber‑sheet analogy: Masses depress a stretchy sheet (spacetime); the steeper the dip, the stronger the “pull.” Orbit as continuous falling: Imagine throwing a ball forward fast enough that Earth’s curvature makes the ground fall away beneath it → a stable orbit. Equivalence principle as an elevator: Inside a sealed, accelerating elevator you cannot tell whether the force you feel is due to gravity or acceleration. --- 🚩 Exceptions & Edge Cases Strong‑field regimes: Near neutron stars or black holes, Newton’s law underestimates precession, time dilation, and light bending. High‑precision Earth navigation: GPS must incorporate both GR (time dilation) and special relativity corrections; Newtonian models alone give meter‑scale errors. Very low accelerations (\(a < 10^{-10}\ \mathrm{m\,s^{-2}}\)): Some alternative theories (MOND) propose deviations from the inverse‑square law. --- 📍 When to Use Which Newtonian calculations – satellites, planetary orbits, escape velocity, engineering problems where \(v \ll c\) and gravitational potentials are weak. GR calculations – perihelion precession, light deflection, gravitational redshift, timing of pulsars, black‑hole dynamics, gravitational‑wave predictions. Dark‑matter modeling – galaxy rotation curves, large‑scale structure simulations; use halo mass profiles (e.g., NFW) instead of only visible mass. MOND or alternative gravity – when testing hypotheses for rotation‑curve anomalies without invoking dark matter. --- 👀 Patterns to Recognize \(1/r^{2}\) dependence shows up in force, field strength, and flux through a spherical surface. Elliptical orbit → Kepler’s laws: (1) Equal areas in equal times, (2) \(T^{2} \propto a^{3}\). GR corrections scale with \((v/c)^{2}\) – look for terms like \(\sim 10^{-9}\) on Earth but larger near compact objects. Gravitational lensing: multiple images or Einstein rings appear when a massive object lies nearly along the line of sight to a background source. --- 🗂️ Exam Traps Confusing “gravity” with “centrifugal force” – the effective weight on Earth is gravity minus centrifugal component; the latter reduces apparent \(g\) at the equator. Using \(v{\text{esc}} = \sqrt{GM/R}\) – missing the factor of 2 leads to a 30 % error. Assuming Newton’s law works for light – light follows null geodesics; only GR predicts its deflection. Treating dark matter as “ordinary matter” – it does not interact electromagnetically; it only contributes to the gravitational potential. Mixing up standard gravitational parameter \(\mu\) and \(g\) – \(\mu = GM\) (units \(\mathrm{m^{3}s^{-2}}\)) is used for orbital mechanics; \(g\) is local acceleration (units \(\mathrm{m\,s^{-2}}\)). ---