# Gravitational Field due to a Point Mass(Edexcel International A Level Physics)

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Katie M

Expertise

Physics

## 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 FG leads to:

• Cancelling mass m, the equation becomes:

• Where:
• g = gravitational field strength due to a point mass M (N kg1)
• 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

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

#### Exam Tip

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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