5.33 Gravitational Field due to a Point Mass

• The gravitational field strength at a point describes how much gravitational force is experienced by a test mass at that point
• The strength of a gravitational field caused by a point mass can be derived using Newton’s Law of Universal Gravitation
  • For calculations involving gravitational forces, a spherical mass can be treated as a point mass at the centre of the sphere

• Newton’s Law of Universal Gravitation states that the magnitude of the attractive gravitational force F between two masses M and m with separation r is equal to:

• The gravitational field strength at a point is defined as the force F per unit mass m

• Substituting the force F with the gravitational force F_G leads to:

• Cancelling mass m, the equation becomes:

• Where:
  • g = gravitational field strength due to a point mass M (N kg^–1)
  • G = Newton’s Gravitational Constant (N m² kg^–2)
  • M = mass of the body producing the gravitational field (kg)
  • r = distance from the centre of mass M  to a chosen point in the field (m)

Step 1: Write the known quantities

• • Mass of Earth = 6.0 × 10²⁴ kg
  • Radius of Earth = 6400 km = 6400 × 10³ m
  • Newton's Universal Gravitation Constant G = 6.67 × 10^-11 N m² kg^–2 
  • Distance from the Earth's surface = 1 million km = (1 × 10⁶) × 10³ m = 1 × 10⁹ m

Step 2: Write the equation for gravitational field strength

• • The gravitational field strength g at some distance r due to a mass M is given by the equation:

g = 

Step 3: Determine the distance r from the centre of mass M

• • The Earth is causing the gravitational field in the question
  • 1 million km from the Earth's surface means the distance from Earth's centre of mass r = (6400 × 10³) + (1 × 10⁹) = 1.0064 × 10⁹ m

Step 4: Substitute values and calculate the gravitational field strength

• • Therefore, the gravitational field strength is given by:

g = = 4.0 × 10^–4 N kg^–1