A LevelPhysicsEdexcelRevision Notes12. Gravitational FieldsGravitational FieldsGravitational Field due to a Point Mass

Gravitational Field due to a Point Mass (Edexcel A Level Physics) : Revision Note

Katie M

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Katie M

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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:

Deriving Gravitational Field Strength (g) equation 1

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

Deriving Gravitational Field Strength (g) equation 2

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

Deriving Gravitational Field Strength (g) equation 3

  • Cancelling mass m, the equation becomes:

Deriving Gravitational Field Strength (g) equation 4

  • 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)

Worked Example

The Earth's gravitational field strength at its surface is 9.81 N kg–1. 

Calculate the Earth's gravitational field strength 1 million km away from its surface. 

Use the following data:

  • Mass of Earth = 6.0 × 1024 kg

  • Radius of Earth = 6400 km

Answer:

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 = fraction numerator G M over denominator r squared end fraction

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 = fraction numerator left parenthesis 6.67 cross times 10 to the power of negative 11 end exponent right parenthesis cross times left parenthesis 6.0 cross times 10 to the power of 24 right parenthesis over denominator left parenthesis 1.0064 cross times 10 to the power of 9 right parenthesis squared end fraction= 4.0 × 10–4 N kg–1

Examiner Tips and Tricks

When using the equation for gravitational field strength, remember that the mass M is the mass causing the gravitational field. The mass m is the object that experiences the gravitational field of M : hence, you may see m referred to as a 'test mass'. 

It should make sense, therefore, that g is defined as the 'force experienced per unit mass' in a gravitational field. The force experienced is by the 'test mass', in the field, m. 

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Katie M

Author: Katie M

Expertise: Physics Content Creator

Katie has always been passionate about the sciences, and completed a degree in Astrophysics at Sheffield University. She decided that she wanted to inspire other young people, so moved to Bristol to complete a PGCE in Secondary Science. She particularly loves creating fun and absorbing materials to help students achieve their exam potential.

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