Gravitation

• Kepler's Laws of Planetary Motion

  • Law of Orbits (1st Law): Planets revolve in elliptical orbits with the Sun at one focus
  • Law of Areas (2nd Law): Areal velocity is constant ($\frac{dA}{dt} = \text{constant}$)
    • Based on Conservation of Angular Momentum
  • Law of Periods (3rd Law): $T^2 \propto a^3$ (Time period squared $\propto$ semi-major axis cubed)

• Universal Law of Gravitation

  • Statement: $F = G \frac{m_1 m_2}{r^2}$
  • Universal Gravitational Constant ($G$):
    • Value: $6.674 \times 10^{-11} \text{ N m}^2/\text{kg}^2$
    • Dimensions: $[M^{-1} L^3 T^{-2}]$
  • Properties: Central force, conservative, obeys inverse-square law

• Acceleration Due to Gravity ($g$)

  • Relation with $G$: $g = \frac{GM}{R^2}$
  • Standard Value: $9.8 \text{ m/s}^2$ at Earth's surface
  • Variations in $g$:
    • Height ($h$): $g' = g \left(1 - \frac{2h}{R}\right)$ (for $h \ll R$)
    • Depth ($d$): $g' = g \left(1 - \frac{d}{R}\right)$
    • At Earth's Center: $g = 0$
    • Latitude ($\phi$) / Rotation: $g' = g - \omega^2 R \cos^2\phi$
    • Shape of Earth: $g_{\text{poles}} > g_{\text{equator}}$ (Earth is flattened at poles)

• Gravitational Field and Potential

  • Gravitational Field Intensity ($E$): Force per unit mass ($E = \frac{F}{m} = \frac{GM}{r^2}$)
  • Gravitational Potential ($V$): Work done per unit mass ($V = -\frac{GM}{r}$)
  • Gravitational Potential Energy ($U$): $U = -\frac{GMm}{r}$
    • Negative Sign: Indicates attractive nature of gravitational force
  • Relation: $E = -\frac{dV}{dr}$

• Escape Velocity ($v_e$)

  • Definition: Minimum projection speed to escape Earth's gravitational pull
  • Formula: $v_e = \sqrt{\frac{2GM}{R}} = \sqrt{2gR}$
  • Value for Earth: $\approx 11.2 \text{ km/s}$
  • Key Feature: Independent of mass and angle of projection of the body

• Earth Satellites

  • Orbital Velocity ($v_o$): $v_o = \sqrt{\frac{GM}{r}} = \sqrt{gR}$ (near surface)
  • Relation between $v_e$ and $v_o$: $v_e = \sqrt{2} \cdot v_o$
  • Time Period ($T$): $T = 2\pi \sqrt{\frac{r^3}{GM}}$
  • Energy of Satellite:
    • Kinetic Energy ($K$): $K = \frac{GMm}{2r}$
    • Potential Energy ($U$): $U = -\frac{GMm}{r}$
    • Total Energy ($E$): $E = -\frac{GMm}{2r}$
    • Binding Energy: $-E = \frac{GMm}{2r}$
  • Types of Satellites:
    • Geostationary Satellite:
      • Period: 24 hours
      • Height: $\approx \mathbf{36,000 \text{ km}}$
      • Direction: West to East (Equatorial plane)
      • Use: Communication
    • Polar Satellite:
      • Period: $\approx \mathbf{84-100 \text{ minutes}}$
      • Height: $\approx \mathbf{500-800 \text{ km}}$
      • Direction: North to South (Polar orbit)
      • Use: Remote sensing & weather forecasting

• Weightlessness

  • Concept: Condition where apparent weight (normal reaction force) becomes zero
  • Conditions:
    • During free fall under gravity
    • Inside an orbiting satellite
    • At the center of Earth