Gravitation

Master Newton's law of gravitation, gravitational field, potential, satellite motion, and Kepler's laws for JEE Physics.

Gravitation describes the force of attraction between masses. It governs planetary motion and satellite orbits.

Overview

graph TD
    A[Gravitation] --> B[Universal Law]
    A --> C[g and its Variation]
    A --> D[Potential & Energy]
    A --> E[Satellites]
    A --> F[Kepler's Laws]

Newton’s Law of Universal Gravitation

$$\boxed{F = G\frac{m_1 m_2}{r^2}}$$

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

Gravitational Field

$$\vec{g} = \frac{\vec{F}}{m} = -\frac{GM}{r^2}\hat{r}$$

At Earth’s surface: $g = \frac{GM}{R^2} \approx 9.8$ m/s²

Variation of g

With Altitude

$$\boxed{g_h = g\left(\frac{R}{R+h}\right)^2 \approx g\left(1 - \frac{2h}{R}\right)}$$

(for h « R)

With Depth

$$\boxed{g_d = g\left(1 - \frac{d}{R}\right)}$$

At center of Earth: $g = 0$

Due to Earth’s Rotation

$$g' = g - \omega^2 R\cos^2\lambda$$

where λ = latitude

  • Maximum g at poles (λ = 90°)
  • Minimum g at equator (λ = 0°)
graph LR
    A[Variation of g] --> B["Altitude: decreases as 1/r²"]
    A --> C["Depth: decreases linearly"]
    A --> D["Latitude: max at poles"]

Gravitational Potential

$$\boxed{V = -\frac{GM}{r}}$$

(Potential at infinity = 0)

Relation with Field

$$g = -\frac{dV}{dr}$$

Gravitational Potential Energy

$$\boxed{U = -\frac{GMm}{r}}$$

Note: U = 0 at infinity, U is negative (bound system)

Escape Velocity

Minimum velocity to escape Earth’s gravitational field:

$$\boxed{v_e = \sqrt{2gR} = \sqrt{\frac{2GM}{R}}}$$

For Earth: $v_e \approx 11.2$ km/s

Properties:

  • Independent of mass of object
  • Independent of direction of projection
  • $v_e = \sqrt{2} \times v_{orbital}$
JEE Tip
Escape velocity is related to orbital velocity by $v_e = \sqrt{2} \cdot v_o$. This is useful for numerical problems.

Satellite Motion

Orbital Velocity

$$\boxed{v_o = \sqrt{\frac{GM}{r}} = \sqrt{gR} \cdot \sqrt{\frac{R}{r}}}$$

For satellite close to Earth: $v_o \approx 7.9$ km/s

Time Period

$$\boxed{T = 2\pi\sqrt{\frac{r^3}{GM}} = 2\pi\sqrt{\frac{r^3}{gR^2}}}$$

Energy of Satellite

$$KE = \frac{1}{2}mv_o^2 = \frac{GMm}{2r}$$ $$PE = -\frac{GMm}{r}$$ $$\boxed{E_{total} = -\frac{GMm}{2r} = \frac{1}{2}PE = -KE}$$

Note: Total energy is negative (bound state)

Geostationary Satellite

  • Time period = 24 hours
  • Height ≈ 36,000 km
  • Orbits in equatorial plane
  • Appears stationary from Earth

Polar Satellite

  • Passes over poles
  • Height ~ 800 km
  • Used for mapping, weather

Kepler’s Laws

1. Law of Orbits

Planets move in elliptical orbits with Sun at one focus.

2. Law of Areas

Line joining planet to Sun sweeps equal areas in equal times.

$$\frac{dA}{dt} = \frac{L}{2m} = \text{constant}$$

This implies conservation of angular momentum.

3. Law of Periods

$$\boxed{T^2 \propto a^3}$$ $$\frac{T^2}{a^3} = \frac{4\pi^2}{GM} = \text{constant}$$

where a = semi-major axis

Common Mistake
In Kepler’s third law, ‘a’ is the semi-major axis, not the radius. For circular orbits, a = r.

Practice Problems

  1. At what height above Earth’s surface is g reduced to g/4?

  2. Calculate escape velocity from a planet with mass 4M_E and radius 2R_E.

  3. A satellite orbits at height equal to Earth’s radius. Find its time period. (R_E = 6400 km, g = 10 m/s²)

  4. Find the ratio of KE to |PE| for a satellite in orbit.

Quick Check
Why do astronauts in orbit experience weightlessness even though gravity is acting on them?

Further Reading