Gravitation Mind Map (JEE)

Cheatsheet Content

🌍 Gravitation 1. Newton’s Law of Gravitation 🍎 ✅ Important Concepts: Force is always attractive, acts along the line joining centers. Applies to point masses; for extended bodies, use integration or consider center of mass for spherical symmetry. 📌 Key Formulas: $F = \frac{G m_1 m_2}{r^2}$ ⚠️ Silly / Common JEE Mistakes: Forgetting $r$ is center-to-center distance, not surface-to-surface. Confusing $G$ (gravitational constant) with $g$ (acceleration due to gravity). 🧠 Concept Traps: Gravitational force between two objects is independent of the medium between them. ⭐ JEE Favourite Facts: Newton's 3rd law pair: $F_{12} = -F_{21}$. $G$ is a universal constant, value $\approx 6.67 \times 10^{-11} \text{ Nm}^2/\text{kg}^2$. 📐 Units / Direction / Nature: Force is a vector. Always attractive. Units: Newtons (N). 2. Gravitational Field & Potential 🌌 ✅ Important Concepts: Gravitational field $\vec{E}$ is force per unit mass. Gravitational potential $V$ is work done per unit mass to bring from infinity. 📌 Key Formulas: $\vec{E} = \frac{\vec{F}}{m_0} = -\frac{GM}{r^2} \hat{r}$ (for point mass) $V = -\frac{GM}{r}$ (for point mass/spherical shell outside) $\vec{E} = -\nabla V$ (field is negative gradient of potential) ⚠️ Silly / Common JEE Mistakes: Sign errors in potential calculations (always negative for attractive force). Confusing field (vector) with potential (scalar). Potential inside a hollow sphere is constant, not zero. 🧠 Concept Traps: Gravitational field inside a uniform solid sphere is proportional to $r$. Potential inside a uniform solid sphere is not constant; it's minimum at center. ⭐ JEE Favourite Facts: At center of a uniform solid sphere: $E=0$, $V = -\frac{3GM}{2R}$. For a uniform spherical shell: $E=0$ inside, $V = -\frac{GM}{R}$ inside. 📐 Units / Direction / Nature: Field: N/kg or m/s$^2$ (vector). Potential: J/kg (scalar). Direction of field is towards the mass creating it. Potential is always negative (reference at infinity). 3. Variation of g (height, depth, latitude) ⛰️ ✅ Important Concepts: $g$ is acceleration due to gravity, effectively the gravitational field strength on Earth's surface. $g$ varies with distance from Earth's center, shape of Earth, and rotation. 📌 Key Formulas: Height $h \ll R_e$: $g_h = g(1 - \frac{2h}{R_e})$ Height $h$ (any): $g_h = \frac{GM}{(R_e+h)^2}$ Depth $d$: $g_d = g(1 - \frac{d}{R_e})$ Latitude $\lambda$: $g_{\lambda} = g - R_e \omega^2 \cos^2 \lambda$ ⚠️ Silly / Common JEE Mistakes: Applying approximation $(1 - \frac{2h}{R_e})$ when $h$ is comparable to $R_e$. Confusing decrease with height (outer sphere) with decrease with depth (inner sphere). Forgetting Earth's rotation effect on $g$ at poles/equator. 🧠 Concept Traps: $g$ is maximum at poles ($\lambda=90^\circ$, $\cos\lambda=0$). $g$ is minimum at equator ($\lambda=0^\circ$, $\cos\lambda=1$). $g$ at center of Earth is zero. ⭐ JEE Favourite Facts: $g$ decreases with both height and depth from surface. The decrease in $g$ for small $h$ and $d$ is approximately $\frac{2h}{R_e}g$ and $\frac{d}{R_e}g$ respectively. For $h=d$, decrease due to height is twice that due to depth. 📐 Units / Direction / Nature: Units: m/s$^2$ (vector, direction towards Earth's center). Scalar magnitude varies with location. 4. Escape Velocity 🚀 ✅ Important Concepts: Minimum velocity required for an object to escape the gravitational field of a planet and never return. Conceptually, kinetic energy + potential energy = 0 at infinity. 📌 Key Formulas: $v_e = \sqrt{\frac{2GM}{R}}$ $v_e = \sqrt{2gR}$ ⚠️ Silly / Common JEE Mistakes: Forgetting the factor of 2 (compare with orbital velocity). Assuming escape velocity depends on the mass of the escaping object (it does not). Using wrong radius (e.g., height above surface instead of $R+h$). 🧠 Concept Traps: Escape velocity is independent of the direction of projection. It's independent of the mass of the projectile. ⭐ JEE Favourite Facts: Escape velocity for Earth $\approx 11.2 \text{ km/s}$. If projected with $v v_e$, it escapes with residual kinetic energy. 📐 Units / Direction / Nature: Units: m/s (scalar magnitude). Direction is irrelevant for magnitude; any direction works if magnitude is achieved. 5. Orbital Motion & Satellites 🛰️ ✅ Important Concepts: Centripetal force for orbit is provided by gravitational force. Kepler's Laws describe planetary motion. 📌 Key Formulas: Orbital velocity: $v_o = \sqrt{\frac{GM}{r}}$ (for circular orbit) Time period: $T = \frac{2\pi r}{v_o} = 2\pi \sqrt{\frac{r^3}{GM}}$ Kepler's 3rd Law: $T^2 \propto r^3$ (where $r$ is semi-major axis) Angular momentum: $L = mvr = \text{constant}$ (Kepler's 2nd Law) ⚠️ Silly / Common JEE Mistakes: Using $R$ instead of $r$ (orbital radius from center of planet). Mixing up $v_o$ and $v_e$. Note $v_e = \sqrt{2} v_o$. Assuming all orbits are circular (most are elliptical). 🧠 Concept Traps: Geosynchronous/Geostationary satellites: $T=24$ hours, orbit above equator. Energy of orbiting satellite is negative and constant for stable orbit. ⭐ JEE Favourite Facts: Kepler's 1st Law: Orbits are ellipses with Sun at one focus. Kepler's 2nd Law: Equal areas swept in equal times (conservation of angular momentum). Binding Energy = $-(\text{Total Energy})$. 📐 Units / Direction / Nature: Velocity: m/s (vector, tangential). Period: seconds. Angular momentum: kg m$^2$/s (vector). 6. Gravitational Energy ⚡ ✅ Important Concepts: Potential energy of a system of masses, defined as work done to assemble them from infinity. Total mechanical energy is sum of kinetic and potential energy. 📌 Key Formulas: Potential Energy (two masses): $U = -\frac{Gm_1 m_2}{r}$ Total Energy of satellite: $E = K + U = \frac{1}{2}mv_o^2 - \frac{GMm}{r} = -\frac{GMm}{2r}$ (for circular orbit) Binding Energy = $-E = \frac{GMm}{2r}$ ⚠️ Silly / Common JEE Mistakes: Sign errors in potential energy (always negative for attractive force). Calculating potential energy for a system of more than two particles: sum of all pairs. Confusing potential $V$ (per unit mass) with potential energy $U$. 🧠 Concept Traps: For an object to escape, its total energy must be $\ge 0$. For a bound (orbiting) system, total energy is negative. ⭐ JEE Favourite Facts: For a circular orbit, $K = -U/2$ and $E = U/2 = -K$. Energy required to move a satellite from orbit $r_1$ to $r_2$: $\Delta E = E_2 - E_1$. 📐 Units / Direction / Nature: Units: Joules (J). Scalar quantity. Energy is negative for bound systems, positive for unbound. 7. Shell Theorem 🥚 ✅ Important Concepts: A uniform spherical shell of matter exerts no net gravitational force on a particle located inside it. A uniform spherical shell of matter attracts a particle outside it as if all the shell's mass were concentrated at its center. 📌 Key Formulas: Outside shell ($r \ge R$): $F = \frac{GMm}{r^2}$, $E = \frac{GM}{r^2}$, $V = -\frac{GM}{r}$ Inside shell ($r ⚠️ Silly / Common JEE Mistakes: Applying shell theorem indiscriminately to non-uniform or non-spherical bodies. Assuming potential inside a shell is zero (it's constant, not zero). Confusing solid sphere results with shell results. 🧠 Concept Traps: The potential inside a hollow sphere is constant and equal to the potential on its surface. This theorem is crucial for calculating gravity inside and outside planets/stars assuming uniform density layers. ⭐ JEE Favourite Facts: This theorem simplifies many complex gravitational problems for spherical symmetry. It's the foundation for understanding $g$ variation with depth. 📐 Units / Direction / Nature: Force (vector), Field (vector), Potential (scalar). Inside a shell, force and field are zero. Potential is constant.