Gravitation

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Gravitation — Key Facts

Newton’s Universal Law of Gravitation: $$F = G\frac{m_1 m_2}{r^2}$$

where $G = 6.67 \times 10^{-11}$ N·m²/kg²

This force is always attractive, acts along the line joining centres, and affects all matter (universal).

Gravitational Field and Acceleration:

On Earth’s surface (M_E = 5.97 × 10²⁴ kg, R_E = 6371 km): $$g = \frac{GM_E}{R_E^2} \approx 9.8 \text{ m/s}^2$$

Orbital Motion:

For satellite in circular orbit: $$v_{orbit} = \sqrt{\frac{GM}{r}} = \sqrt{gr} \approx 7.9 \text{ km/s (near Earth)}$$

Time period of near-Earth orbit: $T = 2\pi\sqrt{\frac{r^3}{GM}} \approx 84$ minutes

Escape Velocity: $$v_e = \sqrt{\frac{2GM}{r}} = \sqrt{2gr} \approx 11.2 \text{ km/s (from Earth)}$$

Note: $v_e = \sqrt{2} \times v_{orbit}$ (for same r)

⚡ JEE Exam Tip: For a satellite in circular orbit, centripetal force = gravitational force: $mv^2/r = GMm/r^2$. This is why orbital speed is independent of satellite mass.

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🟡 Standard — Regular Study (2d–2mo)

For students who want genuine understanding…

Gravitational Field:

Gravitational field strength (acceleration) at distance r: $$g(r) = \frac{GM}{r^2}$$

Gravitational potential: $$V = -\frac{GM}{r}$$

(gravitational potential is always negative, zero at infinity)

Relation between g and V: $$\vec{g} = -\nabla V = -\frac{dV}{dr} \hat{r}$$

Kepler’s Laws:

1. Law of Orbits: All planets move in elliptical orbits with the Sun at one focus.
2. Law of Areas: The line joining a planet to the Sun sweeps equal areas in equal times. (This expresses conservation of angular momentum: L = mvr = constant.)
3. Law of Periods: $T^2 \propto r^3$ (for circular orbits)

From Kepler’s 3rd law: $T^2 = \frac{4\pi^2}{GM}r^3$

Geostationary Satellite:

For a satellite to appear stationary above Earth:

• Orbit must be equatorial (above equator)
• Period = Earth’s rotation period = 24 hours
• Height ≈ 36,000 km above Earth’s surface
• Orbital velocity ≈ 3.1 km/s

⚡ JEE Exam Tip: Polar satellites have lower orbits (~700 km) and faster orbital periods (~100 minutes). They’re used for Earth observation and can cover the entire Earth as it rotates beneath them.

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Gravitational Potential Energy:

Work done in moving mass m from infinity to distance r from M: $$U = -\frac{GMm}{r}$$

Total mechanical energy in orbit: $$E = KE + PE = \frac{1}{2}mv^2 - \frac{GMm}{r} = -\frac{GMm}{2r} = -\frac{KE}{1}$$

(Orbital energy is negative and equals half the potential energy, with opposite sign.)

Variation of g:

With altitude: $g’ = g\left(\frac{R}{R+h}\right)^2$

With depth: $g’ = g\left(1 - \frac{d}{R}\right)$ (assuming uniform Earth density)

At equator: $g_{min}$ due to Earth’s rotation (accounting for centrifugal effect).

Black Holes:

Schwarzschild radius (event horizon): $$R_s = \frac{2GM}{c^2}$$

For Sun: R_s ≈ 3 km; For Earth: R_s ≈ 9 mm

If Earth were compressed to this size, escape speed would equal speed of light.

Gravitational Time Dilation:

Clocks run slower in stronger gravitational fields: $$\frac{dt’}{dt} = \sqrt{1 - \frac{2GM}{rc^2}} = \sqrt{1 - \frac{R_s}{r}}$$

This has been confirmed using GPS satellites (they need to account for both special and general relativistic effects).

Inertial vs Gravitational Mass:

Experimentally, inertial mass = gravitational mass to 1 part in 10¹².

This is the foundation of Einstein’s equivalence principle, which states that gravitational acceleration is indistinguishable from acceleration due to any other cause.

Gravitational Lensing:

Light from distant stars bends around massive objects (like the Sun). $$\theta = \frac{4GM}{c^2 b}$$ (for small deflection angle)

where b = impact parameter (closest approach distance).

This has been observed during solar eclipses and is used to detect exoplanets via gravitational microlensing.

Gravitational Slingshot:

When a spacecraft swings by a planet, it gains or loses speed in the planet’s frame (while the planet loses or gains negligible speed in the Sun’s frame due to its large mass).

$$\Delta v_{spacecraft} \approx 2v_{planet}$$

This technique is essential for missions to the outer solar system.

⚡ JEE Advanced 2022 Analysis: Questions on orbital mechanics, escape velocity, and Kepler’s laws appeared in 2022. For elliptical orbits, remember that speed varies — fastest at perigee (closest to Earth), slowest at apogee. Use conservation of angular momentum (mvr = constant) and energy (E = KE + PE = constant) for these problems.

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