Gravitation

  • Gravitation: A natural phenomenon by which all physical bodies attract each other with a force proportional to their masses and inversely proportional to the square of the distance between them.

1. Newton’s Law of Universal Gravitation

  • Every particle in the universe attracts every other particle.
  • The force is along the line joining their centers.

Formula:

\[ F = G \frac{m_1 m_2}{r^2} \]

  • F: Gravitational Force (attractive always)
  • G: Universal Gravitational Constant
  • m₁, m₂: Masses of the two objects
  • r: Distance between their centers

Key Facts:

  • It is an inverse-square law force.
  • It is a conservative force (work done is path-independent).
  • It is the weakest of the four fundamental forces.
  • It acts even in a vacuum.

2. Acceleration Due to Gravity (g)

The acceleration gained by an object due to gravitational force.

On Earth’s surface:

\[ g = \frac{GM}{R^2} \]

  • M: Mass of Earth
  • R: Radius of Earth

At a height (h) above the surface:

\[ g_h = g \left( \frac{R}{R + h} \right)^2 \]

At a depth (d) below the surface:

\[ g_d = g \left( 1 – \frac{d}{R} \right) \]

Key Facts:

  • Value of ‘g’ decreases with altitude and depth.
  • ‘g’ is maximum at the surface.
  • ‘g’ is zero at the center of the Earth.
  • ‘g’ is slightly greater at the poles than at the equator (due to Earth’s oblate shape and rotation).

3. Gravitational Potential Energy

The energy possessed by an object due to its position in a gravitational field.

For two masses:

\[ U = -G \frac{m_1 m_2}{r} \]

Near Earth’s surface (for small heights):

\[ U = mgh \]

Key Facts:

  • The negative sign signifies a bound system. Energy must be supplied to break the bond.
  • It is maximum (zero) at an infinite distance.

4. Escape Velocity (vesc)

The minimum velocity required for an object to escape the gravitational pull of a planet without any further propulsion.

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

Key Facts:

  • It is independent of the mass of the escaping object.
  • It depends only on the mass and radius of the planet.
  • For Earth: ~11.2 km/s
  • Black Hole: Escape velocity ≥ Speed of Light (c).

5. Kepler’s Laws of Planetary Motion

1. Law of Orbits: All planets move in elliptical orbits with the Sun at one of the two foci.

2. Law of Areas: A line joining a planet and the Sun sweeps out equal areas in equal intervals of time.

  • Implies: A planet moves fastest when it is closest to the Sun (perihelion) and slowest when it is farthest (aphelion).

3. Law of Periods: The square of the orbital period (T) of a planet is directly proportional to the cube of the semi-major axis (a) of its orbit.

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

\[ \frac{T^2}{a^3} = \text{constant} \]

Key Facts:

  • These laws are empirical (based on observation).
  • Newton later proved that these laws are a consequence of his Law of Universal Gravitation.

6. Satellite

An object that revolves around a planet in a stable orbit.

Orbital Velocity (vo): The velocity required to put a satellite into a stable orbit.

\[ v_o = \sqrt{\frac{GM}{r}} \]

  • For a satellite very close to Earth’s surface (r ≈ R): \( v_o = \sqrt{gR} \) ≈ 7.92 km/s

Time Period of a Satellite (T):

\[ T = 2\pi \sqrt{\frac{r^3}{GM}} \]

Geostationary Satellite:

  • Orbits in the equatorial plane.
  • Time period = 24 hours (synchronous with Earth’s rotation).
  • Orbital radius ≈ 42,250 km (from Earth’s center).
  • Height ≈ 36,000 km (from Earth’s surface).
  • Appears stationary from a point on Earth. Used for communication, broadcasting.

Polar Satellite:

  • Orbits over the North and South poles.
  • Height is much lower (~500-800 km).
  • Time period is much shorter (~100 minutes).
  • Covers the entire globe as Earth rotates beneath it. Used for spying, remote sensing, weather forecasting.

7. Weightlessness

  • It is the sensation experienced when the normal reaction force from the ground/surface is zero.
  • It occurs in a freely falling lift or a satellite orbiting Earth.
  • It is not because gravity is zero (gravity still acts), but because the object is in a state of free-fall, canceling the normal force.
  • Astronauts in the ISS are weightless.

Summary of All Formulae

Concept Formula
Newton’s Law \( F = G \frac{m_1 m_2}{r^2} \)
Gravity (Earth’s surface) \( g = \frac{GM}{R^2} \)
Gravity at height (h) \( g_h = g \left( \frac{R}{R + h} \right)^2 \)
Gravity at depth (d) \( g_d = g \left( 1 – \frac{d}{R} \right) \)
Gravitational Potential Energy \( U = -G \frac{m_1 m_2}{r} \)
Escape Velocity \( v_{esc} = \sqrt{\frac{2GM}{R}} = \sqrt{2gR} \)
Orbital Velocity \( v_o = \sqrt{\frac{GM}{r}} \)
Orbital Time Period \( T = 2\pi \sqrt{\frac{r^3}{GM}} \)
Kepler’s 3rd Law \( \frac{T^2}{a^3} = \text{constant} \)

Important Constants & Values

Constant Symbol Value
Universal Gravitational Constant G 6.67430 × 10-11 N m² kg-2
Mass of Earth M 5.972 × 1024 kg
Radius of Earth R 6.371 × 106 m (~6371 km)
Acceleration due to Gravity g 9.8 m s-2 (approx. 10 m s-2 for calculations)
Escape Velocity (Earth) vesc 11.2 km s-1
Orbital Velocity (Near Earth) vo 7.92 km s-1 (or ~8 km/s)
Gravitation