Gravitation Summary

Newton's Law, Gravitational Variation, Orbital Mechanics, and Kepler's Laws

Newton's Law
Gravity Variation
Orbital Mechanics
Kepler's Laws

1. Newton's Law of Gravitation

Newton's Universal Law of Gravitation

The force of interaction between any two particles having masses \( m_1 \) and \( m_2 \) separated by a distance \( r \) is attractive and acts along the line joining the particles.

\[F = \frac{Gm_1 m_2}{r^2}\]

1.1 Key Points

  • G is the universal gravitational constant
  • Numerical value: \( G = 6.67 \times 10^{-11} \, \text{Nm}^2/\text{kg}^2 \)
  • Force is always attractive
  • Force acts along the line joining the centers of masses
  • Follows inverse square law

1.2 Gravitational Constant (G)

Properties:

  • Universal constant
  • Dimension: \([M^{-1}L^3T^{-2}]\)
  • Value: \(6.67 \times 10^{-11} \, \text{Nm}^2/\text{kg}^2\)
  • First measured by Henry Cavendish

2. Variation in Acceleration due to Gravity

Acceleration due to gravity (g) varies with height, depth, and due to Earth's rotation.

With Altitude
\[g' = \frac{g}{\left[1 + \frac{h}{R}\right]^2}\]

For \( h \ll R \):

\[g' = g \left[1 - \frac{2h}{R}\right]\]

Effect: g decreases with height

With Depth
\[g' = g \left[1 - \frac{d}{R}\right]\]

Special cases:

  • At surface: \( g' = g \)
  • At center: \( g' = 0 \)

Effect: g decreases with depth

2.1 Due to Rotation of Earth

\[g' \approx g - \omega^2 R \cos^2 \lambda\]

Where:

  • ω = Angular velocity of Earth
  • R = Radius of Earth
  • λ = Latitude
Location Latitude (λ) Acceleration due to gravity
Equator 0° \( g' \approx g - \omega^2 R \)
Poles 90° \( g' = g \)

Note:

The effective gravity vector \( g' \) is not exactly towards the center of Earth due to centrifugal force.

3. Orbital Mechanics

3.1 Gravitational Potential Energy

Assuming potential energy at infinity to be zero:

\[U = -\frac{GMm}{r}\]

3.2 Mechanical Energy of Orbiting Body

For a satellite of mass m orbiting Earth in circular orbit of radius r:

\[E = \frac{1}{2}mv_0^2 - G\frac{Mm}{r}\]

3.3 Orbital Velocity

\[\frac{mv_0^2}{r} = \frac{GMm}{r^2}\] \[v_0 = \sqrt{\frac{GM}{r}}\]

3.4 Kinetic Energy in Orbit

\[K = \frac{1}{2}mv_0^2 = +G\frac{Mm}{2r}\]

3.5 Total Mechanical Energy

\[E = K + U = G\frac{Mm}{2r} - G\frac{Mm}{r} = -\frac{GMm}{2r}\]

3.6 Escape Speed

Escape Speed

The minimum velocity required to escape from the gravitational field of a planet.

\[v_{es} = \sqrt{\frac{2GM}{R}}\]

Key Points:

  • Escape speed is independent of the mass of the escaping object
  • For Earth: \( v_{es} \approx 11.2 \, \text{km/s} \)
  • Total energy at escape speed is zero

4. Kepler's Laws of Planetary Motion

First Law (Law of Orbits)

All planets move around the sun in elliptical orbits with the sun at one focus.

Second Law (Law of Areas)

The line joining the sun to a planet sweeps out equal areas in equal intervals of time.

Implication:

Planets move faster when closer to the sun and slower when farther away.

Third Law (Law of Periods)

The square of the period of revolution of a planet is proportional to the cube of the semi-major axis of its elliptical orbit.

\[T^2 \propto a^3\] \[T^2 = \kappa a^3\]

Where κ is a constant that applies to all planets.

4.1 Mathematical Relationships

Law Mathematical Expression Physical Significance
First Law \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) Planetary orbits are elliptical
Second Law \(\frac{dA}{dt} = \text{constant}\) Angular momentum conservation
Third Law \(T^2 \propto a^3\) Relates orbital period to distance

Historical Significance:

  • Kepler's laws were derived from Tycho Brahe's observational data
  • They provided evidence against circular orbits and uniform motion
  • Newton later showed they are consequences of his law of gravitation