Gravitation

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Gravitation — Key Facts Newton’s Law: $F = \frac{Gm_1 m_2}{r^2}$; $G = 6.67 \times 10^{-11}$ N·m²/kg² Acceleration due to gravity on Earth: $g = \frac{GM}{r^2} \approx 9.8$ m/s² (varies with altitude, depth, and latitude) Orbital velocity: $v = \sqrt{\frac{GM}{r}} = \sqrt{gr}$ for circular orbit near surface; $v \approx 7.9$ km/s Escape velocity: $v_e = \sqrt{\frac{2GM}{r}} = \sqrt{2gr} \approx 11.2$ km/s from Earth ⚡ Exam tip: For a satellite in circular orbit, gravitational force = centripetal force; $v_{\text{orbit}} = \frac{v_e}{\sqrt{2}}$

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🟡 Standard — Regular Study (2d–2mo)

For students who want genuine understanding of gravitational physics.

Gravitation — CUET Physics Study Guide

Gravitation is a universal force — every mass attracts every other mass. Newton’s Law of Universal Gravitation states that the force of attraction is proportional to the product of masses and inversely proportional to the square of the distance between their centres: $F = \frac{Gm_1 m_2}{r^2}$. The gravitational constant $G = 6.67 \times 10^{-11}$ N·m²/kg² was first measured by Cavendish in 1798 using a torsion balance.

Gravitational Field: The gravitational field strength (acceleration) at a point is $g = \frac{F}{m} = \frac{GM}{r^2}$, directed toward the mass. It is a vector quantity following superposition — for multiple masses, find vector sum of individual fields.

Gravitational Potential: $V = -\frac{GM}{r}$ (negative). It is the work done per unit mass to bring a small test mass from infinity to that point. The negative sign indicates gravitational attraction. Equipotential surfaces are spherical (for point masses) — perpendicular to field lines everywhere.

Kepler’s Laws:

1. All planets move in elliptical orbits with the Sun at one focus (not the centre)
2. The line from Sun to planet sweeps equal areas in equal times — this implies angular momentum conservation ($L = mrv\sin\theta = \text{constant}$)
3. $T^2 \propto r^3$ — for circular orbits, $T^2 = \frac{4\pi^2}{GM}r^3$

Satellites: For a satellite of mass $m$ in circular orbit at radius $r$: orbital speed $v = \sqrt{\frac{GM}{r}}$, orbital period $T = 2\pi\sqrt{\frac{r^3}{GM}}$. A geostationary satellite has $T = 24$ hours and orbits at $h \approx 36,000$ km above Earth’s equator — used for communications. Near-Earth satellites have $T \approx 84$ minutes and $h \approx 200$ km.

Example: Calculate orbital period of a satellite at height $h = 6400$ km (twice Earth’s radius). $r = R_E + h = 6400 + 6400 = 12800$ km $= 1.28 \times 10^7$ m. $T = 2\pi\sqrt{\frac{r^3}{GM}}$ with $M = 6 \times 10^{24}$ kg, $G = 6.67 \times 10^{-11}$. $r^3 \approx 2.10 \times 10^{21}$; $GM = 4 \times 10^{14}$. $T^2 = \frac{4\pi^2 \times 2.10 \times 10^{21}}{4 \times 10^{14}} \approx 2.07 \times 10^7$. $T \approx 4550$ s $\approx 75.8$ minutes.

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Gravitation — Complete CUET Physics Notes

Gravitational Potential Energy: The work done to bring two masses from infinity to distance $r$: $U = -\frac{Gm_1 m_2}{r}$ (negative because we do positive work against gravity to separate them to infinity where $U = 0$). Total mechanical energy of a circular orbit: $E = KE + PE = \frac{GMm}{2r} - \frac{GMm}{r} = -\frac{GMm}{2r} = -\frac{1}{2}KE$. This shows that bound orbits have negative total energy.

Variation of $g$:

• With altitude $h$: $g’ = g\left(\frac{R}{R+h}\right)^2$ (decreases)
• With depth $d$: $g’ = g\left(1 - \frac{d}{R}\right)$ (decreases linearly)
• Due to Earth’s rotation: at equator, $g’ = g - \omega^2 R$ where $\omega = \frac{2\pi}{86400}$ rad/s
• $g$ is maximum at poles (minimum latitude, no rotation effect, slightly oblate Earth)

Black Holes: The Schwarzschild radius (event horizon) is $R_s = \frac{2GM}{c^2}$. For the Sun, $R_s \approx 3$ km; for Earth, $R_s \approx 9$ mm. If Earth were compressed to the size of a pea, it would become a black hole. Nothing, not even light, can escape from inside the Schwarzschild radius.

Gravitational Slingshot: A spacecraft approaching a moving planet can gain or lose speed. In the planet’s frame: spacecraft approaches with $v$, leaves with $-v$ (elastic reflection). In Sun’s frame: speed change $\Delta v = 2v_{\text{planet}}$ in the direction of planet’s motion. Used to reduce fuel requirements for missions to outer planets.

Inertial vs Gravitational Mass: Experimentally, $m_{\text{inertial}} = m_{\text{gravitational}}$ to 1 part in $10^{12}$. This is the foundation of Einstein’s equivalence principle — the basis of general relativity.

Gravitational Binding Energy of Earth: $U = \frac{3}{5}\frac{GM^2}{R} \approx 2.3 \times 10^{32}$ J. This is the energy required to completely disassemble Earth into scattered particles at infinity.

Tidal Forces: The Moon’s gravitational pull on Earth creates a tidal bulge. Spring tides (highest) occur when Sun, Moon, and Earth align (new/full moon); neap tides (lowest) occur at quarter moons when Sun and Moon are perpendicular. The tidal force from the Moon is about twice as strong as from the Sun.

CUET Exam Patterns (2022–2024):

• Kepler’s third law and orbital period calculations are frequent (1–2 marks)
• Escape velocity and orbital velocity relationship ($v_e = \sqrt{2}v_{\text{orbit}}$) is a common MCQ
• Variation of $g$ with altitude and depth appears in Section B
• Gravitational potential energy and satellite energy are occasionally tested
• Common mistakes: confusing mass and weight, forgetting that $g$ varies with location

⚡ Key insight: In orbital mechanics problems, always check whether the satellite is bound (negative energy, elliptical/circular orbit) or unbound (positive energy, hyperbolic trajectory). The relationship $v_{\text{escape}} = \sqrt{2} \times v_{\text{orbital}}$ holds for any circular orbit around the same body. Also, for geostationary orbit, remember it’s always above the equator ($i = 0°$), not any other latitude.

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