Gravitational Force Between Point Masses (Copy)

Point Mass Approximation for Spheres

• For a point outside a uniform spherical object (like a planet or star), the entire mass of the object can be considered to act as if it were concentrated at a point at the centre.
• This is based on Newton’s Shell Theorem, which states:
  • The gravitational effect of a spherically symmetric mass distribution on an external point is identical to that of a point mass located at the centre.
• This assumption greatly simplifies calculations involving planets, moons, and stars.

Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change A2 Level Physics Full Scale Course

Newton’s Law of Universal Gravitation

• The gravitational force F between two point masses m₁ and m₂, separated by distance r, is given by:

  F = Gm₁m₂ / r²

  where:

  • F = gravitational force between the two masses (N)
  • G = gravitational constant = 6.674 × 10⁻¹¹ Nm²/kg²
  • m₁ and m₂ = the two masses (kg)
  • r = distance between the centres of mass (m)
• The force is:
  • Mutual (acts equally and oppositely on both masses)
  • Attractive
  • Acts along the line joining the centres of the two masses
  • Inversely proportional to the square of the distance (1/r²)
  • Directly proportional to the product of the two masses

Applications of Newton’s Law

• Used to calculate:
  • Force between Earth and satellite
  • Force between planets
  • Weight of objects in orbit (gravitational force is the weight in space)

Example

Calculate the force between Earth (mass = 6.0 × 10²⁴ kg) and a 1000 kg satellite 7000 km from Earth’s centre.

F = (6.674 × 10⁻¹¹ × 6.0 × 10²⁴ × 1000) / (7.0 × 10⁶)²
F = (4.0044 × 10¹⁷) / 4.9 × 10¹³ = 8162 N

Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change A2 Level Physics Full Scale Course

Circular Orbits and Gravitational Force as Centripetal Force

• When a satellite or planet is in uniform circular motion around a massive body, the gravitational force acts as the centripetal force required to maintain the orbit.
• Equating the two:

  F = mv² / r (centripetal force)
  F = GmM / r² (gravitational force)

  So:

  mv² / r = GmM / r²

  Cancelling m and simplifying:

  v² = GM / r

  Taking square root:

  v = √(GM / r)

  • where:
    • v = orbital speed
    • r = orbital radius
    • M = mass of the central body (e.g. Earth)
    • G = gravitational constant
• This shows that:
  • Orbital speed is independent of the mass of the orbiting object
  • Higher central mass (M) leads to higher orbital speed
  • Larger orbital radius (r) leads to slower orbital speed

Orbital Period of a Satellite

• The time T for one complete revolution (orbital period) is related to orbital radius as:

  T = 2πr / v

  Substitute v = √(GM / r):

  T = 2πr / √(GM / r)
  T = 2π√(r³ / GM)

  So:

  T² = (4π² / GM) × r³

• This is Kepler’s Third Law: The square of the period is proportional to the cube of the radius.

Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change A2 Level Physics Full Scale Course

Geostationary Orbits

• A geostationary satellite is one that appears stationary above a fixed point on Earth’s surface.
• Also called geosynchronous equatorial orbit.

Key Characteristics:

• Orbital period = 24 hours (same as Earth’s rotation period)
• Direction: West to East (same as Earth’s rotation)
• Located above the equator (0° latitude)
• Orbital radius ≈ 42,000 km from Earth’s centre (approx. 35,786 km above surface)
• Remains fixed relative to one point on the Earth’s surface

Conditions for Geostationary Orbit:

1. The satellite must orbit in the equatorial plane.
2. The satellite must move in the direction of Earth’s rotation (west to east).
3. The time period of orbit = 24 hours.

Advantages of Geostationary Satellites

• Useful for telecommunications, television broadcasting, GPS, and weather monitoring.
• Fixed position means no tracking antenna is needed on Earth.

Disadvantages

• High altitude means:
  • Weak signal strength
  • Long time delay (latency)

Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change A2 Level Physics Full Scale Course

Summary Table