Gravitational Field Of A Point Mass (Copy)

Derivation of g = GM / r²

• Start with Newton’s law of gravitation:

  F = GMm / r²

  where:
  F = gravitational force between two point masses
  G = gravitational constant = 6.674 × 10⁻¹¹ Nm²/kg²
  M = source mass (e.g. Earth)
  m = test mass
  r = distance between centres of mass

• From the definition of gravitational field strength:

  g = F / m

• Substituting Newton’s law into the definition:

  g = (GMm / r²) / m
  g = GM / r²

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

Final Formula

• The gravitational field strength (g) at a distance r from a point mass M is:

  g = GM / r²

  where:
  g = gravitational field strength (N/kg or m/s²)
  G = 6.674 × 10⁻¹¹ Nm²/kg²
  M = mass of the object producing the field (kg)
  r = distance from the centre of the mass (m)

• This equation shows:
  • g is directly proportional to the mass of the object creating the field
  • g is inversely proportional to the square of the distance from the mass

Direction of g

• The gravitational field vector g always points towards the mass M.
• This reflects the attractive nature of gravity.
• The field is radial: field lines extend inward from all directions toward the centre.

Units of g

• From the formula g = GM / r²:
  • G → Nm²/kg²
  • M → kg
  • r² → m²
  • So units of g → (Nm²/kg² × kg) / m² = N/kg
• Since F = ma and g = F / m, then:

  1 N/kg = 1 m/s²

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

Graph of g vs r

• Plotting g against r gives a curve that decreases with the square of the distance.
• g ∝ 1 / r²
• This is a non-linear inverse-square relationship.
• As r increases, g decreases rapidly.

Approximating g as Constant Near Earth’s Surface

• The value of g = GM / r² is accurate for any distance r from Earth’s centre.
• However, near Earth’s surface:
  • Let Earth’s radius = R
  • At a small height h above the surface: r = R + h
  • Since h << R, we can treat r ≈ R (negligible change)
• Therefore:

  g ≈ GM / R² is nearly constant for small heights

• Near Earth’s surface:
  g ≈ 9.81 m/s²
• This simplification allows:
  • Linear equations of motion (SUVAT) to be used with constant g
  • Easier practical calculations in everyday physics

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

Example Calculations

Example 1:
Calculate the gravitational field strength 400 km above Earth’s surface.
Earth’s radius = 6.4 × 10⁶ m
Total distance r = 6.4 × 10⁶ + 4.0 × 10⁵ = 6.8 × 10⁶ m
Earth’s mass = 6.0 × 10²⁴ kg

g = (6.674 × 10⁻¹¹ × 6.0 × 10²⁴) / (6.8 × 10⁶)²
g ≈ 8.67 m/s²

Example 2:
What is the change in g between sea level and 100 m elevation?

• At sea level:
  r = 6.4 × 10⁶ m
  g₁ = GM / r²
• At 100 m:
  r = 6.4 × 10⁶ + 100 = 6.4001 × 10⁶ m
  g₂ = GM / (r + h)²
• Difference is very small, so g is approximately constant at 9.81 m/s².

Summary Table

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