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Gravitation hero

Gravitation

~10 min read

In 30 seconds
  • Newton's Law of Gravitation: F = Gm₁m₂/r². G = 6.674 × 10⁻¹¹ N·m²/kg² (universal constant).
  • g vs G: g = acceleration due to gravity, ≈9.8 m/s² near Earth's surface. G = universal gravitational constant.
  • Escape velocity: From Earth's surface ≈ 11.2 km/s. Orbital velocity (low Earth) ≈ 7.9 km/s.

Newton's law of gravitation explains why apples fall and planets orbit. NDA tests g vs G, escape and orbital velocities, satellites, and Kepler's laws.

Newton's Law of Universal Gravitation

F = Gm₁m₂/r²

  • Every pair of masses attracts each other with a force.
  • Force is proportional to product of masses, inversely proportional to square of distance.
  • G = 6.674 × 10⁻¹¹ N·m²/kg². Universal constant. Measured by Cavendish (1798).
  • Force always attractive.
  • Inverse-square law — doubling distance reduces force to 1/4.

Acceleration Due to Gravity (g)

  • Near Earth's surface: g ≈ 9.8 m/s² (or 9.81, 10 m/s² for approximations).
  • Formula: g = GM/R², where M = mass of Earth, R = radius.
  • Variation:
    • Latitude: g is more at poles (~9.83), less at equator (~9.78) due to Earth's rotation and bulge.
    • Altitude: g decreases with height. At height h, g' = g(R/(R+h))².
    • Depth: g decreases with depth below surface. At depth d, g' = g(1 - d/R).
    • Centre of Earth: g = 0.
  • Weight (W) = mg. Mass is constant; weight varies with location.

Escape and Orbital Velocity

  • Escape velocity (v_e): Minimum velocity required to escape Earth's gravity. v_e = √(2GM/R) = √(2gR) ≈ 11.2 km/s from Earth's surface.
  • Orbital velocity (v_o): Velocity to orbit Earth at altitude h. v_o = √(GM/(R+h)) ≈ 7.9 km/s for low Earth orbit.
  • v_e = √2 × v_o. Escape velocity is about 1.41× orbital velocity.
  • Escape velocity does NOT depend on: mass of projectile, angle of projection.

Satellites

  • Time period of satellite at radius r: T = 2π√(r³/GM).
  • Geostationary satellite: Orbits Earth in 24 hours; appears stationary. Altitude ≈ 36,000 km above equator. Used for communication, weather.
  • Polar satellites: Low Earth orbit; pass over poles. Used for Earth observation, mapping, ISRO Cartosat series.
  • First Indian satellite: Aryabhata (1975, launched by Soviet rocket).
  • India's GSLV, PSLV are launch vehicles. Mangalyaan (2013) reached Mars orbit.

Kepler's Laws

  • 1st (Law of Orbits): All planets orbit the Sun in elliptical orbits with Sun at one focus.
  • 2nd (Law of Areas): Line joining planet and Sun sweeps equal areas in equal times. Planet moves faster when closer to Sun.
  • 3rd (Law of Periods): Square of period ∝ cube of semi-major axis. T² ∝ a³.

Newton's gravitation provided the physics behind Kepler's observed laws.

NDA PYQ Examples

Q: Escape velocity from Earth's surface is approximately:

(a) 7.9 km/s (b) 11.2 km/s (c) 15 km/s (d) 30 km/s

Answer: (b) 11.2 km/s.

Q: Acceleration due to gravity is maximum at:

(a) Equator (b) Poles (c) Centre of Earth (d) Top of Everest

Answer: (b) Poles — Earth is flattened at poles; surface is closer to centre. Also rotation effect.

Q: If radius of Earth shrinks by 1% with mass constant, g would:

(a) Decrease by 1% (b) Decrease by 2% (c) Increase by 1% (d) Increase by 2%

Answer: (d) Increase by 2% — g ∝ 1/R², so dg/g = -2dR/R = +2%.

Q: Geostationary satellite orbital period is:

(a) 1 hour (b) 12 hours (c) 24 hours (d) 30 days

Answer: (c) 24 hours — matches Earth's rotation.

Drill Gravitation for NDA

NDA-pattern items on Gravitation with answer keys and explanations.

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Frequently Asked Questions

Why don't astronauts feel weight in orbit?

Because they (and the space station) are in continuous free fall around Earth. Both fall at the same rate; their relative acceleration is zero. This 'weightlessness' is not absence of gravity — it's free fall.

What is escape velocity?

Minimum speed needed to escape Earth's gravity completely (so the object never returns). For Earth: ~11.2 km/s. Does not depend on the object's mass or launch direction (in vacuum).

Why is g different at equator and poles?

Two effects: (1) Earth's equatorial radius is greater (Earth bulges), so gravity is weaker (further from centre). (2) Centrifugal effect of rotation is maximum at equator, reducing apparent g. Pole g ≈ 9.83 m/s²; equator g ≈ 9.78 m/s².

How do satellites stay in orbit?

Gravity provides the centripetal force needed to keep them moving in circular paths. At orbital velocity, the satellite falls toward Earth at the same rate Earth's surface curves away — it never reaches the surface.

Who discovered the universal law of gravitation?

Isaac Newton (1687). Inspired by the apple story (legend). Generalised earlier work by Galileo, Kepler, and Hooke. Published in Philosophiæ Naturalis Principia Mathematica.