Gravitation (JEE Mains & Adv)

Cheatsheet Content

### Newton's Law of Gravitation - **Statement:** Every particle attracts every other particle with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers. - **Formula:** $$F = G \frac{m_1 m_2}{r^2}$$ - $G$: Universal Gravitational Constant ($6.67 \times 10^{-11} \text{ Nm}^2/\text{kg}^2$) - $m_1, m_2$: Masses of the two particles - $r$: Distance between their centers - **Vector Form:** $$\vec{F}_{12} = -G \frac{m_1 m_2}{r^2} \hat{r}_{12}$$ - $\vec{F}_{12}$: Force on $m_1$ due to $m_2$ - $\hat{r}_{12}$: Unit vector from $m_1$ to $m_2$ (negative sign indicates attractive nature) ### Acceleration Due to Gravity ($g$) - **Definition:** The acceleration experienced by a body due to Earth's gravitational pull. - **On Earth's Surface:** $$g = \frac{GM_E}{R_E^2}$$ - $M_E$: Mass of Earth - $R_E$: Radius of Earth (approx. $9.8 \text{ m/s}^2$) - **Variation with Altitude ($h$):** - For $h \ll R_E$: $$g_h = g \left(1 - \frac{2h}{R_E}\right)$$ - General formula: $$g_h = \frac{GM_E}{(R_E + h)^2}$$ - **Variation with Depth ($d$):** - For $d \ll R_E$: $$g_d = g \left(1 - \frac{d}{R_E}\right)$$ - At center of Earth ($d=R_E$): $g_c = 0$ - **Variation with Latitude ($\lambda$):** Due to Earth's rotation - $$g' = g - R_E \omega^2 \cos^2\lambda$$ - $\omega$: Angular velocity of Earth - At poles ($\lambda = 90^\circ$): $g' = g$ (maximum) - At equator ($\lambda = 0^\circ$): $g' = g - R_E \omega^2$ (minimum) ### Gravitational Field - **Gravitational Field Intensity ($\vec{E}$ or $\vec{I}$):** Force experienced per unit test mass. - $$\vec{E} = \frac{\vec{F}}{m_0}$$ - Units: N/kg or m/s$^2$ - **Due to Point Mass M:** $$E = \frac{GM}{r^2}$$ (towards M) - **Due to Spherical Shell:** - Outside ($r \ge R$): $$E = \frac{GM}{r^2}$$ - Inside ($r ### Gravitational Potential ($V$) - **Definition:** Work done per unit test mass in bringing it from infinity to a point in the gravitational field. - $$V = -\int_{\infty}^{r} \vec{E} \cdot d\vec{r}$$ - Units: J/kg - **Due to Point Mass M:** $$V = -\frac{GM}{r}$$ - **Due to Spherical Shell:** - Outside ($r \ge R$): $$V = -\frac{GM}{r}$$ - Inside ($r ### Gravitational Potential Energy ($U$) - **Definition:** Work done in bringing a mass $m$ from infinity to a point in the gravitational field of mass $M$. - $$U = m V = -\frac{GMm}{r}$$ - Units: Joules (J) - **Change in Potential Energy:** $\Delta U = U_f - U_i = -GMm\left(\frac{1}{r_f} - \frac{1}{r_i}\right)$ - **Potential Energy of a System of n Particles:** Sum of potential energies of all possible pairs. - For three particles $m_1, m_2, m_3$: $$U = -\left(\frac{Gm_1 m_2}{r_{12}} + \frac{Gm_2 m_3}{r_{23}} + \frac{Gm_3 m_1}{r_{31}}\right)$$ ### Escape Velocity ($v_e$) - **Definition:** Minimum velocity required for a body to escape Earth's gravitational field and never return. - **Formula:** $$v_e = \sqrt{\frac{2GM_E}{R_E}} = \sqrt{2gR_E}$$ - For Earth: $v_e \approx 11.2 \text{ km/s}$ - **Relationship with Orbital Velocity:** $v_e = \sqrt{2} v_o$ (for circular orbit close to surface) ### Orbital Velocity ($v_o$) - **Definition:** Velocity required for a satellite to maintain a stable circular orbit around a planet. - **Formula for orbit at height $h$ from surface:** $$v_o = \sqrt{\frac{GM}{(R+h)}}$$ - $M$: Mass of central body - $R$: Radius of central body - **For orbit close to surface ($h \ll R$):** $$v_o = \sqrt{\frac{GM}{R}} = \sqrt{gR}$$ - For Earth: $v_o \approx 7.92 \text{ km/s}$ ### Time Period of Satellite ($T$) - **Formula:** $$T = \frac{2\pi(R+h)}{v_o} = 2\pi \sqrt{\frac{(R+h)^3}{GM}}$$ - **Geostationary Satellite:** - Appears stationary relative to Earth. - $T = 24$ hours. - Orbits in equatorial plane. - Height from Earth's surface: $h \approx 36000 \text{ km}$. ### Energy of an Orbiting Satellite - **Kinetic Energy ($K$):** $$K = \frac{1}{2}mv_o^2 = \frac{1}{2}m\frac{GM}{(R+h)} = \frac{GMm}{2(R+h)}$$ - **Potential Energy ($U$):** $$U = -\frac{GMm}{(R+h)}$$ - **Total Energy ($E$):** $$E = K + U = \frac{GMm}{2(R+h)} - \frac{GMm}{(R+h)} = -\frac{GMm}{2(R+h)}$$ - **Binding Energy:** To remove a satellite from orbit to infinity, energy $+|E|$ must be supplied. - Binding Energy $= -E = \frac{GMm}{2(R+h)}$ ### Kepler's Laws of Planetary Motion - **1. Law of Orbits:** All planets move in elliptical orbits with the Sun at one of the foci. - **2. Law of Areas:** The line joining any planet to the Sun sweeps out equal areas in equal intervals of time. - This implies conservation of angular momentum ($L = \text{constant}$). - Areal velocity $\frac{dA}{dt} = \frac{L}{2m} = \text{constant}$. - **3. Law of Periods:** The square of the time period of revolution of any planet around the Sun is directly proportional to the cube of the semi-major axis of its elliptical orbit. - For circular orbit (radius $r$): $$T^2 \propto r^3 \implies \frac{T^2}{r^3} = \frac{4\pi^2}{GM}$$ - $M$: Mass of the central body (e.g., Sun) ### Important Notes - **Gravitational Force is a Conservative Force:** Work done by gravity is path independent. - **Superposition Principle:** Gravitational force/field/potential due to multiple masses is the vector/scalar sum of individual contributions. - **Weightlessness:** Occurs when the effective acceleration due to gravity is zero (e.g., in a freely falling elevator, or in an orbiting satellite). - **Inertial Mass vs. Gravitational Mass:** Both are equivalent (Principle of Equivalence). - **Centripetal Force in Orbits:** The gravitational force provides the necessary centripetal force for orbital motion. - $$\frac{mv_o^2}{r} = \frac{GMm}{r^2}$$