Class 11 · Chapter 8
Gravitation
Overview, notes, short notes, formula sheet, daily practice problems, previous year questions, and videos for this chapter — all in one place.
Gravitation Short Notes
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Condensed revision points for Gravitation — for quick recall before exams, not a substitute for the full notes.
Newton's Law of Gravitation
- F = Gm₁m₂/r² — always attractive, acts along the line joining the masses.
- G = 6.67 × 10⁻¹¹ N·m²/kg² — universal, same everywhere, independent of medium.
- Valid exactly only for point masses or spherically symmetric bodies.
- Conservative force: work done is path-independent; zero net work over a closed loop.
- Obeys superposition — net force from multiple masses is the vector sum of individual forces.
Gravitational Field Intensity
- I = F/m (force per unit mass), unit N/kg, same dimensions as acceleration.
- Point mass / outside a sphere (r > R): I = GM/r²
- On the surface (r = R): I = GM/R²
- Inside a solid sphere (r < R): I = GMr/R³ — grows linearly, zero at the centre.
- Inside a hollow shell: I = 0 everywhere — no net field inside a shell.
Acceleration Due to Gravity (g) & Its Variation
- g = GM/R² = (4/3)πGRρ ≈ 9.8 m/s² at Earth's surface.
- With height (h << R): gₕ = g(1 − 2h/R) — decreases as 1/r² overall.
- With depth d: g_d = g(1 − d/R) — decreases linearly, zero at the centre.
- g is maximum at the surface; falls off whether you go up or down from it.
- With rotation, at latitude λ: g' = g − Rω²cos²λ — max at poles, min at equator.
- ω for zero apparent weight at equator: ω = √(g/R).
Gravitational PE & Potential
- PE: U = −GMm/r (zero at infinity, negative everywhere else — system is bound).
- Work to raise mass by height h: W = GMmh/[R(R+h)] = mgh/(1 + h/R); → mgh when h << R.
- Potential: V = −GM/r (PE per unit mass). Field-potential link: I = −dV/dr.
- Solid sphere, inside (r < R): V = −(GM/2R³)(3R² − r²); centre value = (3/2) × surface value.
- Hollow shell, inside: V is constant, equal to surface value −GM/R (even though field = 0 there).
- For multi-particle systems, total PE = sum of PE of every pair, no shortcuts.
Escape Velocity & Escape Energy
- v_e = √(2GM/R) = √(2gR) — minimum speed to escape a planet's gravity completely.
- Depends only on the planet (M, R or g, R, or density+R) and launch height — NOT on the object's mass or launch angle.
- Escape energy = GMm/R = (1/2)mv_e² (kinetic energy needed to just reach infinity).
- Earth: v_e ≈ 11.2 km/s. Moon: v_e ≈ 2.3 km/s — too low to retain an atmosphere.
Kepler's Three Laws
- 1st (Orbits): elliptical orbit, Sun at one focus — not the centre.
- 2nd (Areas): equal areas swept in equal time → fastest at perihelion, slowest at aphelion.
- 2nd law ⇔ conservation of angular momentum (gravity is a central force, zero torque about the Sun).
- 3rd (Periods): T² ∝ a³ (semi-major axis); for circular orbit, T² ∝ R³.
Satellite Motion — Velocity, Period & Energy
- Orbital velocity: v₀ = √(GM/r) — depends on central mass & radius only, never on satellite's own mass.
- Close to Earth's surface: v₀ = √(gR) ≈ 7.9 km/s.
- Time period: T = 2π√(r³/GM); near surface, T₀ ≈ 84.6 minutes.
- Larger orbital radius → smaller orbital velocity, longer time period.
- K.E. = GMm/2r, P.E. = −GMm/r, Total energy = −GMm/2r — note |P.E.| = 2 × K.E. always.
- Binding energy = −(Total energy) = GMm/2r = K.E.
- Speed boost to √2 × v₀ (≈ 41.4% increase) → K.E. doubles → satellite escapes.
Geostationary, Polar Satellites & Weightlessness
- Geostationary: equatorial orbit, T = 24 h, height ≈ 36,000 km, v₀ ≈ 3.1 km/s; needs ≥3 satellites for global coverage.
- Polar satellite: ~90° inclination, T ≈ 100 min, height 500–800 km; used for weather/mapping.
- Weightlessness in orbit = continuous free fall, NOT absence of gravity — true weight is still nearly full strength.
- Apparent weight (the part felt as a support force) drops to zero; true weight does not.
Common Exam Traps
- g decreasing with height follows 1/r²; g decreasing with depth follows r — different laws, easy to mix up.
- Escape velocity formula has NO dependence on the projected mass — many options try to sneak 'm' into the answer.
- Inside a shell: field = 0, but potential is NOT zero — it equals the surface value.
- Orbital velocity and time period both depend only on orbital radius and central mass — never on the orbiting satellite's mass.
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